Determine the simplified form of the following product, assuming c and d are non-zero real… — Armed Services Vocational Aptitude Battery (ASVAB)

Determine the simplified form of the following product, assuming $c$ and $d$ are non-zero real numbers:

(2c^{-3}d^5)^{-2}(4c^2d^{-1})

What is the resulting fully simplified expression?

The fully simplified expression is

\frac{c^8}{d^{11}}

The correct answer is obtained by first applying the power of a power rule to

(2c^{-3}d^5)^{-2}

, yielding

\frac{1}{4}c^6d^{-10}

. Multiplying this by

4c^2d^{-1}

involves adding the exponents of like bases, which results in

c^8d^{-11}

. Moving the negative exponent to the denominator yields the fully simplified form.

Apply the power of a product rule to the first term, distributing the

-2

exponent to the coefficient and both variables.

2^{-2}(c^{-3})^{-2}(d^5)^{-2} = \frac{1}{4}c^6d^{-10}

The rules

(xy)^a = x^ay^a

and

(x^a)^b = x^{ab}

require every part of the base to be raised to the outer power.

Multiply the expanded first term by the second term by grouping the coefficients and the like variables together.

(\frac{1}{4} \cdot 4)(c^6 \cdot c^2)(d^{-10} \cdot d^{-1})

Multiplication is commutative, which allows rearranging the terms to combine matching bases.

Multiply the numerical coefficients and add the exponents for the variables with the same base.

1 \cdot c^{6+2} \cdot d^{-10+(-1)} = c^8d^{-11}

The product of powers rule states that when multiplying terms with the same base, their exponents should be added (

x^a \cdot x^b = x^{a+b}

Rewrite the expression so that all exponents are positive.

\frac{c^8}{d^{11}}

A fully simplified algebraic expression is typically written with positive exponents, using the rule

x^{-a} = \frac{1}{x^a}

Operations with Variable Exponents

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