Match each unsimplified algebraic expression on the left with its equivalent, fully simpli… — 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.

$(3a^4b^2)(4a^2b)$ $12a^6b^3$
$\frac{24a^7b^3}{6a^3b^6}$ $4a^4b^{-3}$
$(2a^3b^{-2})^4$ $16a^{12}b^{-8}$
$(5a^0b^3)^{-2}$ $\frac{1}{25b^6}$

Answer

The expressions match as follows: (3a^4b^2)(4a^2b) is equivalent to 12a^6b^3. The quotient (24a^7b^3)/(6a^3b^6) simplifies to 4a^4b^-3. The power (2a^3b^-2)^4 expands to 16a^12b^-8. Finally, (5a^0b^3)^-2 simplifies to 1/(25b^6).

Each expression simplifies directly using the fundamental rules of exponents: product rule (add exponents), quotient rule (subtract exponents), power rule (multiply exponents), and zero/negative exponent rules.

Step-by-Step Solution

Simplify the first expression by multiplying the coefficients and applying the product rule for exponents.

3 * 4 = 12, a^(4+2) = a^6, b^(2+1) = b^3. Combined: 12a^6b^3

When multiplying terms with the same base, add their exponents.

Simplify the second expression by dividing the coefficients and applying the quotient rule for exponents.

24 / 6 = 4, a^(7-3) = a^4, b^(3-6) = b^-3. Combined: 4a^4b^-3

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

Simplify the third expression by applying the power of a product rule.

2^4 = 16, a^(3*4) = a^12, b^(-2*4) = b^-8. Combined: 16a^12b^-8

When raising a power to a power, multiply the exponents. Apply the exponent to the numerical coefficient as well.

Simplify the fourth expression by applying the zero exponent rule, then the power rule.

a^0 = 1. 5^-2 = 1/25. b^(3*-2) = b^-6. Combined: 1/(25b^6)

Any non-zero term raised to the power of 0 is 1. Negative exponents indicate taking the reciprocal.

Key Concept

Applying the product, quotient, power, zero, and negative rules of exponents to simplify algebraic expressions with multiple variables and coefficients.

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