If a = -27, b = -\frac{1}{4}, and c = -2, what is the value of the algebraic expression a^… — ACT

If $a = -27$ , $b = -\frac{1}{4}$ , and $c = -2$ , what is the value of the algebraic expression $a^{-4/3} - b^{-2} \cdot c^{-5}$ ?

The correct value of the algebraic expression is

\frac{83}{162}

Evaluating each term of the expression with the given values yields:

a^{-4/3} = \frac{1}{81}

b^{-2} = 16

, and

c^{-5} = -\frac{1}{32}

. Applying the order of operations, we multiply

16

-\frac{1}{32}

first to obtain

-\frac{1}{2}

. We then subtract this product from

\frac{1}{81}

, which simplifies to

\frac{1}{81} + \frac{1}{2} = \frac{83}{162}

Substitute the given values of

a

b

, and

c

into the algebraic expression.

The expression is written as

(-27)^{-4/3} - (-\frac{1}{4})^{-2} \cdot (-2)^{-5}

This establishes the numerical expression to be evaluated.

Evaluate the first term,

(-27)^{-4/3}

(-27)^{-4/3} = \frac{1}{(-27)^{4/3}} = \frac{1}{((-27)^{1/3})^4} = \frac{1}{(-3)^4} = \frac{1}{81}

A negative exponent indicates a reciprocal, and a fractional exponent of

4/3

indicates taking the cube root and then raising to the fourth power.

Evaluate the second term,

(-\frac{1}{4})^{-2}

(-\frac{1}{4})^{-2} = (-4)^2 = 16

Raising a fraction to a negative integer power is equivalent to raising its reciprocal to the corresponding positive integer power.

Evaluate the third term,

(-2)^{-5}

(-2)^{-5} = \frac{1}{(-2)^5} = -\frac{1}{32}

Evaluating a negative base raised to an odd negative power results in a negative unit fraction.

Substitute the evaluated terms back into the original expression and apply the order of operations.

E = \frac{1}{81} - 16 \cdot \left(-\frac{1}{32}\right) = \frac{1}{81} - \left(-\frac{16}{32}\right) = \frac{1}{81} + \frac{1}{2} = \frac{2 + 81}{162} = \frac{83}{162}

Multiplication must be performed before subtraction according to standard mathematical order of operations.

Evaluating expressions containing multiple variables with fractional exponents, negative bases, and standard order of operations.

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