Assume x and y are non-zero real numbers. When the algebraic expression \frac{(2x^{-3}y^4)… — Armed Services Vocational Aptitude Battery (ASVAB)

Assume

x

and

y

are non-zero real numbers. When the algebraic expression

\frac{(2x^{-3}y^4)^3}{4x^5y^{-2}}

is fully simplified to the form

cx^ay^b

, what is the integer value of

b

Answer: 14

Answer

The expression simplifies to

2x^{-14}y^{14}

. Because the question asks for the exponent of

y

, which is denoted by

b

in the target expression

cx^ay^b

, the correct value is 14.

Step-by-Step Solution

Apply the power of a product rule to the numerator.

(2x^{-3}y^4)^3 = 2^3(x^{-3})^3(y^4)^3 = 8x^{-9}y^{12}

When an algebraic term with coefficients and variables is raised to a power, the outer exponent must be distributed to each individual factor inside the parentheses.

Set up the division by applying the quotient rule for each base.

\frac{8x^{-9}y^{12}}{4x^5y^{-2}} = \left(\frac{8}{4}\right) \cdot \left(\frac{x^{-9}}{x^5}\right) \cdot \left(\frac{y^{12}}{y^{-2}}\right)

Separating the expression by identical bases allows the independent application of the quotient rule for exponents.

Evaluate the exponent for the

y

term to find the value of

b

y^{12} \div y^{-2} = y^{12 - (-2)} = y^{12 + 2} = y^{14}

When dividing exponential terms with the same base, subtract the denominator's exponent from the numerator's exponent. Care must be taken with the double negative.

Identify the requested integer value.

b = 14

The question specifically asks for the integer value of

b

in the final simplified form

cx^ay^b

, which represents the exponent of the

y

term.

Key Concept

Applying the power of a power and quotient of powers rules for variable exponents.

Question

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