Precalculus Review: Exponents, Radicals, Factoring, Rational Expressions, and Equations

Q1. Simplify the exponential expression: $(-5x^3y^{-4})(2x^{-1}y)$

Background

Topic: Properties of Exponents

This question tests your ability to apply the rules of exponents, including multiplying powers with the same base and handling negative exponents.

Key Terms and Formulas:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Negative Exponent: $a^{-n} = \frac{1}{a^n}$

Step-by-Step Guidance

1. Distribute the multiplication across the coefficients and variables: $(-5 \cdot 2)(x^3 \cdot x^{-1})(y^{-4} \cdot y^1)$

2. Multiply the coefficients: $-5 \cdot 2$

3. Apply the product of powers property to $x$ and $y$ terms: $x^{3 + (-1)}$ and $y^{-4 + 1}$

4. Combine the results to form a single expression. If you have negative exponents, rewrite them as fractions.

Try solving on your own before revealing the answer!

Final Answer: $-10x^2y^{-3}$ or $\displaystyle -10x^2/ y^3$

We multiplied the coefficients, added the exponents for like bases, and rewrote negative exponents as reciprocals.

Q2. Simplify: $\dfrac{2^{-8}x^{-1}4}{2^{-5}x^{-4}y^{8}}$

Background

Topic: Exponent Rules and Simplifying Rational Expressions

This question tests your ability to simplify expressions with exponents, especially when dividing powers with the same base and handling negative exponents.

Key Terms and Formulas:

• Quotient of Powers: $a^m / a^n = a^{m-n}$

• Negative Exponent: $a^{-n} = 1/a^n$

Step-by-Step Guidance

1. Rewrite $4$as $2^2$ to have all terms with base $2$.

2. Combine the exponents for $2$in the numerator: $2^{-8} \cdot 2^2$.

3. Apply the quotient of powers property for $2$and$x$ terms.

4. Simplify the $y$ term, noting that $y^8$ is only in the denominator.

Try solving on your own before revealing the answer!

Final Answer: $2^{-1}x^{3}y^{-8}$ or $\displaystyle \frac{x^3}{2y^8}$

We combined exponents for each base and rewrote negative exponents as reciprocals.

Q3. Simplify: $\left(\dfrac{x^5y^0}{x^6y}\right)^{-2}$

Background

Topic: Exponent Rules, Negative and Zero Exponents

This question tests your understanding of negative exponents, zero exponents, and simplifying complex fractions with exponents.

Key Terms and Formulas:

• Zero Exponent: $a^0 = 1$ (for $a \neq 0$)

• Negative Exponent: $a^{-n} = 1/a^n$

• Power of a Quotient: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$

Step-by-Step Guidance

1. Simplify $y^0$ to $1$ in the numerator.

2. Combine the exponents for $x$ and $y$ in the fraction: $x^{5-6}$ and $y^{0-1}$.

3. Apply the negative exponent to the entire fraction: $\left(\dfrac{x^{-1}}{y^{-1}}\right)^{-2}$.

4. Use the property $\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n$ to rewrite the expression.

Try solving on your own before revealing the answer!

Final Answer: $x^2y^2$

We simplified the exponents and applied the negative exponent rule to invert and square the fraction.

Q4. Simplify: $\dfrac{17.08 \times 10^{-8}}{4 \times 10^5}$

Background

Topic: Scientific Notation and Exponent Rules

This question tests your ability to divide numbers in scientific notation and apply exponent rules.

Key Terms and Formulas:

• Scientific Notation: $a \times 10^n$

• Quotient of Powers: $10^a / 10^b = 10^{a-b}$

Step-by-Step Guidance

1. Divide the coefficients: $17.08 / 4$.

2. Apply the exponent rule to the powers of $10$: $10^{-8} / 10^5$.

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

Try solving on your own before revealing the answer!

Final Answer: $4.27 \times 10^{-13}$

We divided the coefficients and subtracted the exponents for the powers of ten.

Q5. Add or subtract: $\sqrt{3x} + 7\sqrt{48x} - 3\sqrt{108x}$

Background

Topic: Simplifying Radical Expressions

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

Key Terms and Formulas:

• Simplifying Radicals: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

• Like Radicals: Terms with the same radicand can be combined.

Step-by-Step Guidance

1. Simplify each radical by factoring out perfect squares.

2. Rewrite each term in the form $a\sqrt{3x}$ if possible.

3. Combine like terms by adding or subtracting their coefficients.

Try solving on your own before revealing the answer!

Final Answer: $-2\sqrt{3x}$

After simplifying each radical and combining like terms, the result is $-2\sqrt{3x}$.

Q6. Simplify: $y\sqrt[3]{54x} - \sqrt[3]{250xy^3}$

Background

Topic: Simplifying Cube Roots and Radical Expressions

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

Key Terms and Formulas:

• Cube Root: $\sqrt[3]{a^3} = a$

• Simplifying Cube Roots: Factor out perfect cubes from under the radical.

Step-by-Step Guidance

1. Factor $54x$ and $250xy^3$ to identify perfect cubes.

2. Simplify each cube root by extracting perfect cubes.

3. Multiply the simplified radical by $y$ where needed.

4. Combine like terms if possible.

Try solving on your own before revealing the answer!

Final Answer: $3y\sqrt[3]{2x} - 5y\sqrt[3]{2x}$ or $-2y\sqrt[3]{2x}$

We simplified each cube root and combined like terms.

Q7. Rationalize the denominator: $\dfrac{\sqrt{6}}{\sqrt{17} + 2}$

Background

Topic: Rationalizing Denominators

This question tests your ability to eliminate radicals from the denominator by multiplying by the conjugate.

Key Terms and Formulas:

• Conjugate: For $a + b$, the conjugate is $a - b$.

• Rationalizing: Multiply numerator and denominator by the conjugate of the denominator.

Step-by-Step Guidance

1. Identify the conjugate of the denominator: $\sqrt{17} - 2$.

2. Multiply both numerator and denominator by this conjugate.

3. Expand the denominator using $(a+b)(a-b) = a^2 - b^2$.

4. Simplify the numerator and denominator as much as possible.

Try solving on your own before revealing the answer!

Final Answer: $\dfrac{\sqrt{6}(\sqrt{17} - 2)}{17 - 4}$ or $\dfrac{\sqrt{6}(\sqrt{17} - 2)}{13}$

We multiplied by the conjugate and simplified the denominator.

Q8. Rationalize the denominator: $\dfrac{2}{\sqrt{5} + \sqrt{7}}$

Background

Topic: Rationalizing Denominators with Two Terms

This question tests your ability to rationalize denominators with two radical terms by multiplying by the conjugate.

Key Terms and Formulas:

• Conjugate: $\sqrt{5} - \sqrt{7}$

• Difference of Squares: $(a+b)(a-b) = a^2 - b^2$

Step-by-Step Guidance

1. Multiply numerator and denominator by the conjugate $\sqrt{5} - \sqrt{7}$.

2. Expand the denominator using the difference of squares formula.

3. Simplify the numerator and denominator.

Try solving on your own before revealing the answer!

Final Answer: $\dfrac{2(\sqrt{5} - \sqrt{7})}{5 - 7}$ or $\dfrac{2(\sqrt{5} - \sqrt{7})}{-2}$

We multiplied by the conjugate and simplified the denominator.

Q9. Simplify using properties of exponents: $(6x^{1/3})(3x^{3/2})$

Background

Topic: Properties of Exponents

This question tests your ability to multiply expressions with fractional exponents.

Key Terms and Formulas:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

Step-by-Step Guidance

1. Multiply the coefficients: $6 \cdot 3$.

2. Add the exponents for $x$: $x^{1/3} \cdot x^{3/2} = x^{1/3 + 3/2}$.

3. Find a common denominator to add the exponents.

Try solving on your own before revealing the answer!

Final Answer: $18x^{11/6}$

We multiplied the coefficients and added the exponents using a common denominator.

Q10. Simplify: $\dfrac{100x^{2/3}}{10x^{1/4}}$

Background

Topic: Exponent Rules and Simplifying Rational Expressions

This question tests your ability to divide powers with the same base and simplify coefficients.

Key Terms and Formulas:

• Quotient of Powers: $a^m / a^n = a^{m-n}$

Step-by-Step Guidance

1. Divide the coefficients: $100 / 10$.

2. Subtract the exponents for $x$: $x^{2/3 - 1/4}$.

3. Find a common denominator to subtract the exponents.

Try solving on your own before revealing the answer!

Final Answer: $10x^{5/12}$

We divided the coefficients and subtracted the exponents using a common denominator.

Q11. Factor completely: $4x^2 - 49$

Background

Topic: Factoring the Difference of Squares

This question tests your ability to recognize and factor expressions of the form $a^2 - b^2$.

Key Terms and Formulas:

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

Step-by-Step Guidance

1. Recognize $4x^2$ as $(2x)^2$ and $49$as $7^2$.

2. Apply the difference of squares formula.

Try solving on your own before revealing the answer!

Final Answer: $(2x - 7)(2x + 7)$

We factored the expression using the difference of squares formula.

Q12. Factor completely: $27x^3 + 64$

Background

Topic: Factoring Sums of Cubes

This question tests your ability to factor expressions of the form $a^3 + b^3$.

Key Terms and Formulas:

• Sum of Cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Step-by-Step Guidance

1. Recognize $27x^3$ as $(3x)^3$ and $64$as $4^3$.

2. Apply the sum of cubes formula.

Try solving on your own before revealing the answer!

Final Answer: $(3x + 4)(9x^2 - 12x + 16)$

We factored the sum of cubes using the standard formula.