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Chapter 0: Problem 124

What difference is there in simplifying \(\sqrt[3]{(-5)^{3}}\) and \(\sqrt[4]{(-5)^{4} ?}\)

Short Answer

Expert verified
The difference is that the simplified result of the cube root \(\sqrt[3]{(-5)^{3}}\) is -5 and the simplified result of the square root \(\sqrt[4]{(-5)^{4}}\) is 5.

Step by step solution

01

Recognize the root types

The first expression involves a cube root (odd-degree root) while the second is a square root (even-degree root). This is important because the root degree affects how we handle negative numbers. For an odd-degree root of a negative number, the result is negative. For an even-degree root of a negative number, the result does not exist in the real number world because there's no real number that you can square (multiply by itself) to get a negative number.
02

Simplify the cube root

For the first expression \(\sqrt[3]{(-5)^{3}}\), simplify it to \( -5 \). This is because when we raise -5 to an exponent of 3, we get -125. Then, a cube root of -125 is -5.
03

Simplify the square root

For the second expression \(\sqrt[4]{(-5)^{4}}\), simplify it to 5. This is because when we raise -5 to an exponent of 4, we get 625. Then, a square root of 625 is 5.

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