If p = -3, q = \frac{1}{2}, and r = -27, what is the value of the algebraic expression bel… — ACT

p = -3

q = \frac{1}{2}

, and

r = -27

, what is the value of the algebraic expression below?

\frac{p^2 q^{-3} + r^{1/3}}{(2pq + 5)^2 + q^{-1}}

Answer: 11.5

11.5

Evaluating each component of the expression step-by-step using order of operations:
1. Numerator:

p^2 q^{-3} + r^{1/3} = (-3)^2 \left(\frac{1}{2}\right)^{-3} + (-27)^{1/3} = 9 \cdot 8 + (-3) = 72 - 3 = 69

.
2. Denominator:

(2pq + 5)^2 + q^{-1} = \left(2(-3)\left(\frac{1}{2}\right) + 5\right)^2 + \left(\frac{1}{2}\right)^{-1} = (-3 + 5)^2 + 2 = 2^2 + 2 = 4 + 2 = 6

.
3. Final Quotient:

\frac{69}{6} = 11.5

Evaluate the terms in the numerator

Numerator = 69

Since

p = -3

p^2 = 9

. With

q = \frac{1}{2}

q^{-3} = 2^3 = 8

. Thus,

p^2 q^{-3} = 9 \cdot 8 = 72

. Also,

r^{1/3} = (-27)^{1/3} = -3

. Adding these values gives

72 + (-3) = 69

Evaluate the terms in the denominator

Denominator = 6

First,

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

. Then

(2pq + 5)^2 = (-3 + 5)^2 = 2^2 = 4

. Next,

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

. Adding these components gives

4 + 2 = 6

Divide the numerator by the denominator

11.5

Dividing the numerator (69) by the denominator (6) yields

\frac{69}{6} = 11.5

(or

\frac{23}{2}

Evaluating algebraic expressions with negative bases, negative exponents, fractional exponents, and order of operations.

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