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

Match each unsimplified algebraic expression on the left with its fully simplified equivalent form on the right. Assume $p$ and $q$ represent non-zero numbers.

$(4p^5 q^{-2})(2p^{-1} q^6)$ $8p^4 q^4$
$\frac{15p^7 q^2}{3p^2 q^{-3}}$ $5p^5 q^5$
$(3p^3 q^2)^2$ $9p^6 q^4$
$\frac{(2p^2 q^3)^3}{4p^4 q^7}$ $2p^2 q^2$

Answer

The correct matches are: (4p^5 q^-2)(2p^-1 q^6) matches 8p^4 q^4; the fraction 15p^7 q^2 over 3p^2 q^-3 matches 5p^5 q^5; (3p^3 q^2)^2 matches 9p^6 q^4; and the fraction (2p^2 q^3)^3 over 4p^4 q^7 matches 2p^2 q^2.

Each expression simplifies to exactly one of the given target forms by consistently applying the laws of exponents (product, quotient, and power rules) and handling numerical coefficients accurately.

Step-by-Step Solution

Simplify the first expression using the product rule for exponents.

The coefficients multiply to 8. Adding the exponents for p gives 5 + (-1) = 4. Adding the exponents for q gives -2 + 6 = 4. The result is 8p^4 q^4.

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

Simplify the second expression using the quotient rule for exponents.

The coefficients divide to 5. Subtracting the p exponents gives 7 - 2 = 5. Subtracting the q exponents gives 2 - (-3) = 5. The result is 5p^5 q^5.

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

Simplify the third expression using the power of a product rule.

The coefficient squared is 3^2 = 9. Multiplying the p exponent by 2 gives 3 * 2 = 6. Multiplying the q exponent by 2 gives 2 * 2 = 4. The result is 9p^6 q^4.

When raising a power to a power, multiply the exponents. Remember to also apply the power to the numerical coefficient.

Simplify the fourth expression by applying the power rule first, then the quotient rule.

The numerator becomes 8p^6 q^9. Dividing this by 4p^4 q^7 gives a coefficient of 2, p exponent of 6 - 4 = 2, and q exponent of 9 - 7 = 2. The result is 2p^2 q^2.

Order of operations requires handling the exponent on the entire numerator before performing division.

Key Concept

Applying the product, quotient, and power rules of exponents to simplify algebraic expressions.

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