If m = -4 and n = -\frac{1}{2}, what is the value of the algebraic expression \frac{m^2 - … — ACT

If $m = -4$ and $n = -\frac{1}{2}$ , what is the value of the algebraic expression $\frac{m^2 - 4n^3}{(m + 2n)^2}$ ?

\frac{33}{50}

Substituting

m = -4

and

n = -\frac{1}{2}

into the numerator gives

(-4)^2 - 4\left(-\frac{1}{2}\right)^3 = 16 - \left(-\frac{1}{2}\right) = \frac{33}{2}

. Substituting into the denominator gives

\left(-4 + 2\left(-\frac{1}{2}\right)\right)^2 = (-5)^2 = 25

. Dividing the numerator by the denominator yields

\frac{33/2}{25} = \frac{33}{50}

Evaluate the terms in the numerator:

m^2

and

4n^3

m^2 = (-4)^2 = 16

and

4n^3 = 4\left(-\frac{1}{2}\right)^3 = 4\left(-\frac{1}{8}\right) = -\frac{1}{2}

Substitute

m = -4

and

n = -\frac{1}{2}

while applying powers before multiplication according to PEMDAS.

Calculate the complete numerator by subtracting the evaluated terms.

m^2 - 4n^3 = 16 - \left(-\frac{1}{2}\right) = 16 + \frac{1}{2} = \frac{33}{2}

Subtracting a negative quantity is equivalent to adding its positive counterpart.

Evaluate the expression inside the denominator parentheses, then square it.

m + 2n = -4 + 2\left(-\frac{1}{2}\right) = -4 - 1 = -5

, and

(-5)^2 = 25

Grouped operations inside parentheses must be evaluated prior to applying outer exponents.

Divide the numerator by the denominator to get the final simplified fraction.

\frac{\frac{33}{2}}{25} = \frac{33}{2 \times 25} = \frac{33}{50}

Dividing a fraction by an integer combines the denominators.

Evaluating algebraic expressions involving negative bases, fractions, and order of operations

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