Match each unsimplified algebraic expression on the left with its equivalent fully simplif… — Armed Services Vocational Aptitude Battery (ASVAB)

Match each unsimplified algebraic expression on the left with its equivalent fully simplified form on the right. Assume all variables represent non-zero real numbers.

$(3a^4b^2)(-4a^2b^{-5})$ $\frac{-12a^6}{b^3}$
$\frac{16a^3b^6}{8a^7b^2}$ $\frac{2b^4}{a^4}$
$(2a^3b^5)^4$ $16a^{12}b^{20}$
$(5a^{-2}b^0)^{-2}$ $\frac{a^4}{25}$

Answer

The product rule matches to -12a^6/b^3, the quotient rule matches to 2b^4/a^4, the power of a product rule matches to 16a^12b^20, and the negative exponent simplification matches to a^4/25.

Matching each pair correctly requires correctly identifying and applying four distinct fundamental laws of exponents: the product rule (adding exponents), the quotient rule (subtracting exponents), the power of a product rule (multiplying exponents), and the rules for evaluating zero and negative exponents.

Step-by-Step Solution

Simplify

(3a^4b^2)(-4a^2b^{-5})

using the product rule for exponents.

-12a^6b^{-3}

, which simplifies to

\frac{-12a^6}{b^3}

When multiplying expressions with the same base, add their exponents and multiply their numerical coefficients.

Simplify

\frac{16a^3b^6}{8a^7b^2}

using the quotient rule for exponents.

2a^{-4}b^4

, which simplifies to

\frac{2b^4}{a^4}

When dividing expressions with the same base, subtract the denominator's exponent from the numerator's exponent and divide the numerical coefficients.

Simplify

(2a^3b^5)^4

using the power of a product rule.

16a^{12}b^{20}

When raising a product to a power, raise the numerical coefficient to that power and multiply each variable's exponent by the outer power.

Simplify

(5a^{-2}b^0)^{-2}

using the zero and negative exponent rules.

First,

b^0 = 1

. Then evaluate

5^{-2}a^4

, which simplifies to

\frac{a^4}{25}

Any non-zero term raised to the zero power equals 1. Negative exponents indicate the reciprocal of the base.

Key Concept

Operations with Variable Exponents

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic