Q1. Simplify:

Background

Topic: Order of Operations (PEMDAS/BODMAS)

This question tests your ability to simplify expressions by applying the correct order of operations, including distributing negatives and combining like terms.

Key Terms and Formulas:

• Order of Operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

• Distributive Property:

Step-by-Step Guidance

1. Start by simplifying inside the innermost parentheses: .

2. Subtract from 3: .

3. Distribute the to each term inside the brackets: .

4. Combine the result with the 7 outside the brackets.

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Q2. Rewrite the expression without absolute value bars. Don’t use decimal numbers.

Background

Topic: Absolute Value Expressions

This question tests your understanding of how to express an absolute value algebraically, considering different cases based on the value of the variable.

Key Terms and Formulas:

• Absolute Value:

Step-by-Step Guidance

1. Recall the definition of absolute value for an expression .

2. Set and consider the two cases: and .

3. Write the piecewise expression for based on these cases.

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Q3. Write an equivalent expression without a fraction and simplify:

Background

Topic: Simplifying Rational Expressions

This question tests your ability to multiply and divide monomials, apply exponent rules, and simplify the result.

Key Terms and Formulas:

• Product of Powers:

• Quotient of Powers:

Step-by-Step Guidance

1. Multiply the numerators: .

2. Simplify the product in the numerator by combining like terms.

3. Divide the result by .

4. Apply the exponent rules to simplify the expression, ensuring all exponents are positive.

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Q4. Write an equivalent expression with positive exponents and simplify: and determine all real numbers that must be excluded from the domain.

Background

Topic: Exponent Rules and Domain Restrictions

This question tests your ability to simplify expressions with exponents and to identify values that make the denominator zero (domain restrictions).

Key Terms and Formulas:

• Product of Powers:

• Negative Exponent:

• Domain: Set of all real numbers for which the expression is defined (denominator ≠ 0)

Step-by-Step Guidance

1. Combine the exponents in the numerator: .

2. Apply the quotient rule for exponents to combine numerator and denominator.

3. Rewrite any negative exponents as positive exponents.

4. Identify values of and that would make the denominator zero.

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Q5. Write in decimal notation:

Background

Topic: Scientific Notation to Decimal Notation

This question tests your ability to convert a number from scientific notation to standard decimal form.

Key Terms and Formulas:

• Scientific Notation: means move the decimal point places to the right (if ).

Step-by-Step Guidance

1. Identify the exponent on 10, which tells you how many places to move the decimal point.

2. Move the decimal point in six places to the right.

3. Fill in zeros as needed to complete the number.

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Q6. Write in scientific notation:

Background

Topic: Decimal Notation to Scientific Notation

This question tests your ability to express a number in the form where .

Key Terms and Formulas:

• Scientific Notation:

• is a number between 1 and 10, is an integer

Step-by-Step Guidance

1. Move the decimal point in so that the new number is between and .

2. Count how many places you moved the decimal to the left.

3. Write the number in the form with the correct sign for .

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Q7. Perform the indicated computation and write the answer in scientific notation:

Background

Topic: Multiplying Numbers in Scientific Notation

This question tests your ability to multiply numbers in scientific notation and express the result in proper scientific notation.

Key Terms and Formulas:

• Multiplication:

Step-by-Step Guidance

1. Multiply the coefficients: .

2. Add the exponents: .

3. Combine the results to write the answer in scientific notation.

4. If necessary, adjust the coefficient so it is between 1 and 10, and adjust the exponent accordingly.

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Q8. Simplify:

Background

Topic: Properties of Radicals

This question tests your ability to multiply square roots and simplify the result.

Key Terms and Formulas:

• Product Property:

• Simplifying Radicals: Factor out perfect squares if possible

Step-by-Step Guidance

1. Multiply the expressions under the radicals: .

2. Write the product under a single square root.

3. Simplify the expression under the radical, and factor out perfect squares if possible.

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Q9. Write an equivalent expression without a fraction and simplify:

Background

Topic: Simplifying Rational Expressions

This question tests your ability to divide coefficients and simplify exponents.

Key Terms and Formulas:

• Division: if and are numbers

• Exponent Rules: remains unchanged if not divided by another term

Step-by-Step Guidance

1. Divide the coefficients: .

2. Write the result with the term.

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Q10. Subtract square roots:

Background

Topic: Simplifying Radical Expressions

This question tests your ability to simplify square roots and combine like terms.

Key Terms and Formulas:

• Simplifying Radicals:

• Combine like terms if the radical part is the same

Step-by-Step Guidance

1. Simplify and by factoring out perfect squares.

2. Multiply the simplified radicals by their coefficients ($6).

3. Check if the resulting terms have like radicals and can be combined.

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Q11. Rationalize the denominator in

Background

Topic: Rationalizing Denominators

This question tests your ability to eliminate square roots from the denominator of a fraction.

Key Terms and Formulas:

• Rationalizing: Multiply numerator and denominator by a value that will eliminate the radical in the denominator

• Simplifying Radicals:

Step-by-Step Guidance

1. Simplify as much as possible.

2. Multiply numerator and denominator by the appropriate radical to rationalize the denominator.

3. Simplify the resulting expression.

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Q12. Subtract:

Background

Topic: Operations with Fractions

This question tests your ability to subtract fractions with different denominators.

Key Terms and Formulas:

• Common Denominator: Find the least common denominator (LCD) to combine fractions

• Subtract numerators after converting to common denominator

Step-by-Step Guidance

1. Find the least common denominator (LCD) for 8 and 3.

2. Rewrite each fraction with the LCD as the denominator.

3. Subtract the numerators and write the result over the common denominator.

4. Simplify the fraction if possible.

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Q13. Simplify with higher roots:

Background

Topic: Radical Expressions and Roots

This question tests your understanding of cube roots and their properties.

Key Terms and Formulas:

• Cube Root: is the number that, when cubed, gives

• Radical to Exponent:

Step-by-Step Guidance

1. Express the cube root as a rational exponent.

2. Simplify if possible, or leave in exponent form if no further simplification is possible.

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Q14. Simplify with rational exponents:

Background

Topic: Rational Exponents and Exponent Rules

This question tests your ability to apply exponent rules with rational exponents and simplify the result.

Key Terms and Formulas:

• Power of a Power:

• Product of Powers:

• Rational Exponents:

Step-by-Step Guidance

1. Apply the power rule to .

2. Rewrite as .

3. Multiply the results, combining like terms using exponent rules.

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Q15. Add the polynomials:

Background

Topic: Polynomial Addition

This question tests your ability to combine like terms when adding polynomials.

Key Terms and Formulas:

• Like Terms: Terms with the same variable and exponent

• Add coefficients of like terms

Step-by-Step Guidance

1. Group like terms from both polynomials (e.g., , , , constants).

2. Add the coefficients for each group.

3. Write the resulting polynomial in standard form (descending powers).

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Q16. Multiply the polynomials and combine like terms:

Background

Topic: Polynomial Multiplication and Simplification

This question tests your ability to distribute and combine like terms in polynomial expressions.

Key Terms and Formulas:

• Distributive Property:

• Combine like terms after multiplication

Step-by-Step Guidance

1. Distribute to each term inside the parentheses.

2. Write out each product.

3. Combine like terms to simplify the expression.

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Q17. Factor:

Background

Topic: Factoring Polynomials

This question tests your ability to factor out the greatest common factor (GCF) from a polynomial.

Key Terms and Formulas:

• Greatest Common Factor (GCF): The largest expression that divides each term

• Factoring:

Step-by-Step Guidance

1. Identify the GCF of the coefficients and variables in both terms.

2. Factor the GCF out of each term.

3. Write the expression as a product of the GCF and the remaining terms.

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Q18. Factor by grouping:

Background

Topic: Factoring by Grouping

This question tests your ability to factor a four-term polynomial by grouping terms and factoring common factors.

Key Terms and Formulas:

• Factoring by Grouping: Group terms to factor out common factors, then factor the common binomial.

Step-by-Step Guidance

1. Group the terms into two pairs: and .

2. Factor out the GCF from each group.

3. Look for a common binomial factor in both groups.

4. Factor out the common binomial.

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Q19. Factor the trinomial:

Background

Topic: Factoring Trinomials

This question tests your ability to factor a quadratic trinomial into two binomials.

Key Terms and Formulas:

• Factoring Quadratics: where and multiply to and add to

Step-by-Step Guidance

1. Identify two numbers that multiply to and add to $2$.

2. Write the trinomial as a product of two binomials using these numbers.

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Q20. Factor:

Background

Topic: Difference of Squares

This question tests your ability to recognize and factor a difference of squares.

Key Terms and Formulas:

• Difference of Squares:

Step-by-Step Guidance

1. Recognize $49 and as .

2. Apply the difference of squares formula.

Try solving on your own before revealing the answer!