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Chapter 7: Problem 105

Simplify. Assume that \(x \geq 0 .\) \(\sqrt[4]{25}\)

Short Answer

Expert verified
\root{4}{25} = \root{2}{5}

Step by step solution

01

Understand the Problem

The given problem requires simplifying the fourth root of 25, which is \(\root{4}{25}\).
02

Rewrite as Exponential Form

Express the fourth root in terms of exponents. \(\root{4}{25}\) can be written as \(25^{1/4}\).
03

Simplify the Exponent

To simplify \(25^{1/4}\), rewrite 25 as a power of a smaller base. Notice that \(25 = 5^2\). So, \(25^{1/4} = (5^2)^{1/4}\).
04

Apply Exponent Rules

Use the rule \( (a^m)^n = a^{mn} \). Here, \((5^2)^{1/4} = 5^{2 * 1/4} = 5^{1/2}\).
05

Convert Back to Radicals

The expression \(5^{1/2}\) can be rewritten as the square root of 5, \(\root{2}{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fourth Roots
To simplify fourth roots, it's helpful to understand what they are. The fourth root of a number is the value that, when multiplied by itself four times, equals the original number. For example, the fourth root of 16 is 2 because \(2 \times 2 \times 2 \times 2 = 16\). To find a fourth root, we often start by expressing the number in exponential form. This conversion can make calculations simpler and help when applying various exponent rules. Throughout this process, remember to handle radicals and exponents carefully, as small mistakes can lead to big errors later on.
Exponent Rules
Exponent rules are essential for simplifying expressions involving exponents. One key rule is \(a^m \times a^n = a^{m+n}\). This rule shows that when you multiply like bases, you add the exponents. Similarly, \( \frac{a^m}{a^n} = a^{m-n} \), demonstrates dividing like bases means subtracting the exponents. For our specific problem, the rule \((a^m)^n = a^{mn}\) is crucial. Applying it helps in simplifying expressions like \((5^2)^{1/4}\). Each of these rules plays a vital role in converting, simplifying, and understanding exponential forms, making them fundamental tools in mathematics. Keep these rules handy, and practice using them to build a strong foundation in algebra.
Rewriting Exponents
Rewriting exponents can simplify complex expressions. It often involves breaking down numbers into smaller, more manageable pieces. For instance, in the problem above, we rewrite 25 as \(5^2\) to make the simplification process easier. This step is particularly useful when the base has a clear and simple exponent form. Rewriting might also mean converting between radical and exponential forms. For example, \(\sqrt[4]{25}\) converts to \(25^{1/4}\). Using smaller bases or simpler forms helps apply exponent rules effectively. Practicing this skill will enhance your mathematical problem-solving ability, leading to more accurate and efficient solutions.

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