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Math  /  Numbers & Operations

QuestionSimplify the expression. −2783\sqrt[3]{\frac{-27}{8}}
Enter the exact answer. Hint: You can write roots as fractional exponents, for example −2783\sqrt[3]{\frac{-27}{8}} as (−27/8)∧(1/3)(-27 / 8)^{\wedge}(1 / 3). However, the answer to this question is a fraction without needing a rowt symbol.

Studdy Solution

STEP 1

1. The given expression is a cube root, which can be expressed as a fractional exponent.
2. The cube root of a fraction can be simplified by taking the cube root of the numerator and the denominator separately.
3. The final answer should be in the form of a simplified fraction without any root symbols.

STEP 2

1. Rewrite the cube root as a fractional exponent.
2. Simplify the fractional exponent by separating the numerator and denominator.
3. Calculate the cube root of both the numerator and the denominator.
4. Combine the results to get the simplified fraction.

STEP 3

Rewrite the cube root as a fractional exponent. −2783=(−278)13 \sqrt[3]{\frac{-27}{8}} = \left( \frac{-27}{8} \right)^{\frac{1}{3}}

STEP 4

Separate the fractional exponent to apply to both the numerator and the denominator individually. (−278)13=(−27)13813 \left( \frac{-27}{8} \right)^{\frac{1}{3}} = \frac{(-27)^{\frac{1}{3}}}{8^{\frac{1}{3}}}

STEP 5

Calculate the cube root of the numerator, −27-27. (−27)13=−3 (-27)^{\frac{1}{3}} = -3

STEP 6

Calculate the cube root of the denominator, 88. 813=2 8^{\frac{1}{3}} = 2

STEP 7

Combine the results to get the simplified fraction. (−27)13813=−32 \frac{(-27)^{\frac{1}{3}}}{8^{\frac{1}{3}}} = \frac{-3}{2}
The simplified expression is: −2783=−32 \sqrt[3]{\frac{-27}{8}} = \frac{-3}{2}

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