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The cube root is an essential concept in mathematics that deals with finding a number that, when multiplied by itself three times, gives the original number. When you see a cube root written as \( \sqrt[3]{x} \), it is asking: "What number, when cubed, will result in \( x \)?"

In our exercise, we dealt with \( \sqrt[3]{(-5)^{3}} \). Firstly, we resolve \((-5)^3\), which is the same as multiplying \(-5\) by itself three times. The calculation is as follows:

• \( -5 \times -5 \times -5 = -125 \)

Now, we take the cube root of \(-125\). Finding the cube root means seeking a number which, when cubed, results in \(-125\). Since \( -5 \times -5 \times -5 = -125 \), the cube root of \(-125\) is \(-5\).

This example shows that for cube roots, negative numbers remain negative because each negative sign is neutralized within the power cycle of three. Therefore, \( \sqrt[3]{(-5)^{3}} \) simplifies to \(-5\).

The fourth root is similar to the cube root, but we seek a number that, when multiplied by itself four times, results in the original number. It's written as \( \sqrt[4]{x} \). In the exercise, \( \sqrt[4]{(-5)^{4}} \) involves the fourth power of \(-5\). Raising \(-5\) to the fourth power means:

• Multiply \(-5\) four times: \( (-5) \times (-5) \times (-5) \times (-5) = 625 \)

Now, we need to find the fourth root of 625. If we multiply \(5\) by itself four times:

• \( 5 \times 5 \times 5 \times 5 = 625 \)

Thus, the fourth root of 625 is \(5\).

Unlike the cube root, the fourth root of a negative number raised to an even power gives a positive result. This occurs because multiplying a negative number by itself an even number of times results in a positive number. Hence the simplification of \( \sqrt[4]{(-5)^{4}} \) is positive \(5\).

Simplifying expressions with roots of powers often involves recognizing whether the exponent and the root index are odd or even. This recognition helps to determine the sign of the result.

Here's a simple guide:

• If you have an odd power root (like a cube root), a negative inside stays negative because multiplying an odd number of negatives results in a negative.
• If you have an even power root (like a fourth root), negative bases become positive. Multiplying an even number of negatives results in a positive number.

In our exercise:

• The expression \( \sqrt[3]{(-5)^{3}} \) simplifies to \(-5\) because the cube root of a negative number powered to an odd number remains negative.
• The expression \( \sqrt[4]{(-5)^{4}} \) simplifies to \(5\) because the fourth root of a negative number raised to an even power results in a positive.

Understanding this basic principle can greatly help in simplifying complex expressions, as it allows one to predict the outcome simply by noting the nature of the root and the power.