{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Simplify the exponential expression: "},{"type":"inlineMath","attrs":{"latex":"(-5x^3y^{-4})(2x^{-1}y)"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Properties of Exponents"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to apply exponent rules, including multiplication of powers, negative exponents, and combining like terms."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Product of powers: "},{"type":"inlineMath","attrs":{"latex":"a^m \\cdot a^n = a^{m+n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative exponent: "},{"type":"inlineMath","attrs":{"latex":"a^{-n} = \\frac{1}{a^n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying coefficients and variables separately"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the coefficients: "},{"type":"inlineMath","attrs":{"latex":"-5 \\times 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" terms: "},{"type":"inlineMath","attrs":{"latex":"x^3 \\times x^{-1}"}},{"type":"text","text":" using the product of powers rule."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" terms: "},{"type":"inlineMath","attrs":{"latex":"y^{-4} \\times y^1"}},{"type":"text","text":" using the product of powers rule."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the expression with simplified exponents, but do not calculate the final numeric value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"-10x^2y^{-3}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying coefficients gives "},{"type":"inlineMath","attrs":{"latex":"-10"}},{"type":"text","text":", combining exponents for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" gives "},{"type":"inlineMath","attrs":{"latex":"x^{3+(-1)} = x^2"}},{"type":"text","text":", and for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" gives "},{"type":"inlineMath","attrs":{"latex":"y^{-4+1} = y^{-3}"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Alternatively, you can write "},{"type":"inlineMath","attrs":{"latex":"y^{-3}"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{y^3}"}},{"type":"text","text":" if required."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\frac{2^{-8}x^{-1}4}{2^{-5}x^{-4}y^{8}}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Properties of Exponents and Simplifying Fractions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to simplify expressions with exponents in both the numerator and denominator, including negative exponents."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient of powers: "},{"type":"inlineMath","attrs":{"latex":"a^m / a^n = a^{m-n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative exponent: "},{"type":"inlineMath","attrs":{"latex":"a^{-n} = \\frac{1}{a^n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying and dividing coefficients"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the numerator and denominator, grouping like bases together."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the quotient rule for each base: $2"},{"type":"inlineMath","attrs":{"latex":", "}},{"type":"text","text":"x"},{"type":"inlineMath","attrs":{"latex":", and "}},{"type":"text","text":"y$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the exponents for each variable and coefficient."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the result with positive exponents, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"2^{-3}x^{3}4y^{-8}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After applying exponent rules, you get "},{"type":"inlineMath","attrs":{"latex":"2^{-8-(-5)} = 2^{-3}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x^{-1-(-4)} = x^{3}"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"y^{0-8} = y^{-8}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\left(\\frac{x^6y}{x^6y}\\right)^{-2}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Properties of Exponents and Powers of Quotients"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of simplifying quotients and applying negative exponents to the entire expression."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient rule: "},{"type":"inlineMath","attrs":{"latex":"\\frac{a^m}{a^n} = a^{m-n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Power of a quotient: "},{"type":"inlineMath","attrs":{"latex":"\\left(\\frac{a}{b}\\right)^n = \\frac{a^n}{b^n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative exponent: "},{"type":"inlineMath","attrs":{"latex":"a^{-n} = \\frac{1}{a^n}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the fraction inside the parentheses."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the negative exponent to the simplified fraction."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the expression using exponent rules, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: $1$"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The fraction simplifies to $1"},{"type":"inlineMath","attrs":{"latex":", and any power of $1"}},{"type":"text","text":" is $1$."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\frac{17.08 \\times 10^{-8}}{4 \\times 10^5}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Scientific Notation and Properties of Exponents"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to divide numbers in scientific notation and apply exponent rules."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Scientific notation: "},{"type":"inlineMath","attrs":{"latex":"a \\times 10^n"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient rule for exponents: "},{"type":"inlineMath","attrs":{"latex":"10^m / 10^n = 10^{m-n}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide the coefficients: "},{"type":"inlineMath","attrs":{"latex":"17.08 / 4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the exponent rule: "},{"type":"inlineMath","attrs":{"latex":"10^{-8} / 10^5"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the results to write the answer in scientific notation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"4.27 \\times 10^{-13}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Dividing the coefficients gives "},{"type":"inlineMath","attrs":{"latex":"4.27"}},{"type":"text","text":", and subtracting exponents gives "},{"type":"inlineMath","attrs":{"latex":"10^{-8-5} = 10^{-13}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Add or subtract: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3x} + 7\\sqrt{48x} - 3\\sqrt{108x}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Simplifying Radicals and Combining Like Terms"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to simplify radical expressions and combine like terms."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplifying radicals: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{ab} = \\sqrt{a} \\cdot \\sqrt{b}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combining like terms: Only terms with the same radical part can be combined."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify each radical term by factoring out perfect squares."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify which terms have the same radical part."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine like terms, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3x} + 42\\sqrt{3x} - 9\\sqrt{3x} = 34\\sqrt{3x}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"All terms simplify to multiples of "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{3x}"}},{"type":"text","text":", so you can combine them."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Subtract: "},{"type":"inlineMath","attrs":{"latex":"y\\sqrt[3]{54x} - \\sqrt[3]{250xy^3}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Simplifying Cube Roots and Combining Like Terms"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to simplify cube roots and combine like terms."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplifying cube roots: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt[3]{ab} = \\sqrt[3]{a} \\cdot \\sqrt[3]{b}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combining like terms: Only terms with the same cube root part can be combined."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify each cube root by factoring out perfect cubes."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite each term in its simplest form."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if the cube root parts are like terms and combine if possible, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"3y\\sqrt[3]{2x} - 5\\sqrt[3]{5x}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each cube root is simplified and the terms are not like terms, so they cannot be combined further."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Rationalize the denominator: "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\sqrt{6}}{\\sqrt{17} + 2}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rationalizing Denominators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to eliminate radicals from the denominator by multiplying by the conjugate."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Conjugate: For "},{"type":"inlineMath","attrs":{"latex":"a + b"}},{"type":"text","text":", the conjugate is "},{"type":"inlineMath","attrs":{"latex":"a - b"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying by the conjugate: "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{a+b} \\cdot \\frac{a-b}{a-b}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the conjugate of the denominator: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{17} - 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply numerator and denominator by the conjugate."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the denominator using the difference of squares formula, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\sqrt{6}(\\sqrt{17} - 2)}{17 - 4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying by the conjugate eliminates the radical in the denominator."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Rationalize the denominator: "},{"type":"inlineMath","attrs":{"latex":"\\frac{2}{\\sqrt{5} + \\sqrt{7}}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rationalizing Denominators with Two Radicals"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to rationalize denominators by multiplying by the conjugate when there are two radicals."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Conjugate: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{5} - \\sqrt{7}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Difference of squares: "},{"type":"inlineMath","attrs":{"latex":"(a+b)(a-b) = a^2 - b^2"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply numerator and denominator by the conjugate "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{5} - \\sqrt{7}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the denominator using the difference of squares formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand the numerator, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{2(\\sqrt{5} - \\sqrt{7})}{5 - 7} = \\frac{2(\\sqrt{5} - \\sqrt{7})}{-2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying by the conjugate eliminates the radicals in the denominator."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Simplify using properties of exponents: "},{"type":"inlineMath","attrs":{"latex":"(6x^{1/3})(3x^{3/2})"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Properties of Exponents"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to multiply expressions with fractional exponents."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Product of powers: "},{"type":"inlineMath","attrs":{"latex":"a^m \\cdot a^n = a^{m+n}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiplying coefficients"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the coefficients: "},{"type":"inlineMath","attrs":{"latex":"6 \\times 3"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the exponents for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"x^{1/3} \\times x^{3/2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the result as a single term, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"18x^{\\frac{1}{3} + \\frac{3}{2}} = 18x^{\\frac{11}{6}}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Adding the exponents gives "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{3} + \\frac{3}{2} = \\frac{11}{6}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. 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Simplify the rational expression: "},{"type":"inlineMath","attrs":{"latex":"\\frac{3x + 2}{15x^2 + 22x + 8}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Simplifying Rational Expressions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to factor the denominator and simplify rational expressions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factoring quadratics"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplifying fractions"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor the denominator "},{"type":"inlineMath","attrs":{"latex":"15x^2 + 22x + 8"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check for common factors in the numerator and denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the fraction, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{3x + 2}{(3x + 2)(5x + 4)} = \\frac{1}{5x + 4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The denominator factors as "},{"type":"inlineMath","attrs":{"latex":"(3x + 2)(5x + 4)"}},{"type":"text","text":", so the numerator cancels."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. 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