If x = -2, y = \frac{1}{3}, and z = -4, what is the value of the algebraic expression \fra… — ACT

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

\frac{42}{5}

Substituting

x = -2

y = \frac{1}{3}

, and

z = -4

into the expression yields

(-2)^3 = -8

(\frac{1}{3})^{-1} = 3

, and

(-2 - (-4))^2 = 2^2 = 4

. Thus, the numerator equals

-8 \times 3 - 4 = -28

. The denominator equals

-2 + (\frac{1}{3})(-4) = -2 - \frac{4}{3} = -\frac{10}{3}

. Dividing the numerator by the denominator gives

\frac{-28}{-\frac{10}{3}} = \frac{84}{10} = \frac{42}{5}

Substitute the given values into the numerator of the expression:

x^3 y^{-1} - (x - z)^2

Numerator =

(-2)^3 \left(\frac{1}{3}\right)^{-1} - (-2 - (-4))^2 = (-8)(3) - (2)^2 = -24 - 4 = -28

First apply exponent rules and basic arithmetic inside the parentheses following standard order of operations.

Substitute the given values into the denominator of the expression:

x + y z

Denominator =

-2 + \left(\frac{1}{3}\right)(-4) = -2 - \frac{4}{3} = -\frac{6}{3} - \frac{4}{3} = -\frac{10}{3}

Multiply

y

and

z

first, then find a common denominator to add the fraction to the integer.

Divide the simplified numerator by the simplified denominator.

\frac{-28}{-\frac{10}{3}} = -28 \times \left(-\frac{3}{10}\right) = \frac{84}{10} = \frac{42}{5}.

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Evaluating Algebraic Expressions with Negative Numbers and Negative Exponents

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