If p = -3, q = -\frac{1}{2}, and r = 8, what is the value of the algebraic expression belo… — ACT

If $p = -3$ , $q = -\frac{1}{2}$ , and $r = 8$ , what is the value of the algebraic expression below?

\frac{p^2 q^{-3} - r^{4/3}}{\left(pq - \frac{1}{2}\right)^2 - r^{1/3}}

Answer:The value of the expression is 【88】.

Evaluating each component using exponent rules and standard order of operations yields

-72 - 16 = -88

in the numerator and

1 - 2 = -1

in the denominator. Dividing

-88

-1

results in the final value of

88

Evaluate the terms in the numerator containing negative and rational exponents.

p^2 q^{-3} = (-3)^2 \left(-\frac{1}{2}\right)^{-3} = 9 \cdot (-8) = -72

and

r^{4/3} = 8^{4/3} = (8^{1/3})^4 = 2^4 = 16

Apply exponent rules for negative bases with integer and fractional powers:

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

and

a^{m/n} = (\sqrt[n]{a})^m

Subtract the evaluated terms to determine the total numerator value.

\text{Numerator} = -72 - 16 = -88.

Combine the evaluated terms according to the numerator expression

p^2 q^{-3} - r^{4/3}

Evaluate the grouped and exponential terms in the denominator.

pq - \frac{1}{2} = (-3)\left(-\frac{1}{2}\right) - \frac{1}{2} = \frac{3}{2} - \frac{1}{2} = 1

. Then

\left(pq - \frac{1}{2}\right)^2 = 1^2 = 1

, and

r^{1/3} = 8^{1/3} = 2

Follow the order of operations by simplifying inside the parentheses first, then applying the exponent.

Calculate the denominator and divide the numerator by the denominator.

\text{Denominator} = 1 - 2 = -1

. Dividing the numerator by the denominator gives

\frac{-88}{-1} = 88$.

Perform the final subtraction in the denominator and divide to simplify the fraction fully.

Evaluating Algebraic Expressions with Negative and Fractional Exponents

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