Match each unsimplified algebraic expression with its fully simplified equivalent form. As… — Armed Services Vocational Aptitude Battery (ASVAB)

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

$(3x^{-2}y^4)(5x^6y^{-3})$ $15x^4y$
$\frac{30x^5y^{-2}}{2x^9y^3}$ $\frac{15}{x^4y^5}$
$(-2x^3y^{-1})^4$ $\frac{16x^{12}}{y^4}$
$\left(\frac{3x^{-2}}{y^4}\right)^{-2}$ $\frac{x^4y^8}{9}$

Answer

Each unsimplified expression matches exactly one fully simplified form by applying the product, quotient, and power rules of exponents.

The correct matches are found by systematically applying the fundamental properties of exponents: adding exponents for multiplication, subtracting for division, multiplying for powers of powers, and converting negative exponents to reciprocals.

Step-by-Step Solution

Simplify the product of powers expression: (3x^-2y^4)(5x^6y^-3)

15x^4y

Multiply numerical coefficients together. Add exponents for variables with the same base according to the product rule.

Simplify the quotient of powers expression: (30x^5y^-2) / (2x^9y^3)

15 / (x^4y^5)

Divide numerical coefficients. Subtract the denominator's exponent from the numerator's exponent. Convert negative exponents to positive by moving them to the denominator.

Simplify the power of a product expression: (-2x^3y^-1)^4

(16x^12) / y^4

Raise every factor inside the parentheses to the 4th power. Note that an even power of a negative coefficient yields a positive result.

Simplify the expression with an outer negative exponent: (3x^-2 / y^4)^-2

(x^4y^8) / 9

Multiply all inner exponents by the outer exponent -2. Then take the reciprocal of terms with negative exponents to rewrite them with positive exponents.

Key Concept

Operations with Variable Exponents

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Step-by-Step Solution

Key Concept

Common Mistakes

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