Problem 124 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplification of \(\sqrt[3]{(-5)^{3}}\) is -5, whereas the simplification of \(\sqrt[4]{(-5)^{4}}\) is 5. This is because the cube root of a number cubed yields the original number, even if it is negative, and the fourth root of a number to the power of four yields the absolute value of the original number.

Step 1: Simplify the Cube Root

To simplify \(\sqrt[3]{(-5)^{3}}\), we use the fact that the cube root of a number raised to the power of 3 is just the original number. This means that \(\sqrt[3]{(-5)^{3}} = -5\).

Step 2: Simplify the Fourth Root

To simplify \(\sqrt[4]{(-5)^{4}}\), we again use the properties of roots and exponents. However, because we're dealing with the fourth root (an even root) of a number raised to an even power, the result will be the absolute value of the original number. So, \(\sqrt[4]{(-5)^{4}} = | -5 | = 5\).

Step 3: Compare

We can now clearly see the difference between these two expressions. While the cube root of a negative number raised to the power of 3 gives back the original negative number, the fourth root of that same number raised to the power of 4 yields its absolute value, thus turning it to a positive number.