Problem 107 \(\sqrt[5]{32}\)... [FREE SOLUTION]

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Chapter 10: Problem 107

\(\sqrt[5]{32}\)

Short Answer

Expert verified

Step by step solution

Identify the problem

The goal is to find the fifth root of 32, which can be written as \(\root[5]{32} \).

Rewrite 32 as a power of 2

Recognize that 32 can be written as \(2^5\). This is because \(2 \times 2 \times 2 \times 2 \times 2 = 32\).

Apply the fifth root

Since \(2^5 = 32\), taking the fifth root of both sides gives \( \root[5]{2^5} = \root[5]{32}\).

Simplify the expression

Using the properties of exponents and roots, \(\root[5]{2^5}\) simplifies to 2.

Conclusion

Based on the simplification, the fifth root of 32 is 2.

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Key Concepts

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

Radicals

In mathematics, a radical is a symbol that indicates the root of a number. The root can be square roots, cube roots, or in this case, the fifth root. Radicals can be written in two main ways:

Using the radical symbol like \(\root[5]{32}\)
Or, using fractional exponents (more on this later)

The radical expression \(\root[5]{32}\) is asking us to find a number which, when multiplied by itself five times, gives 32. This specific radical is called the 'fifth root'. Calculating radicals often involves recognizing number patterns or using prime factorization to simplify.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. The expression \(2^5\) means that 2 is multiplied by itself 5 times:
\[ 2 \times 2 \times 2 \times 2 \times 2 = 32 \] When dealing with radicals, it's useful to rewrite the number under the radical as a power. For example, we can rewrite 32 as \(2^5\). To break it down:

32 = 2 * 2 * 2 * 2 * 2 = \(2^5\)

This step is crucial as it makes applying the radical simpler.
Exponents and roots are closely related. The fifth root is the opposite operation of raising a number to the power of 5. Thus, \( \root[5]{2^5}\) simplifies to 2.

Simplification

Simplification is the process of making an expression easier to understand or solve. When simplifying the radical \( \root[5]{32}\), we follow these steps:

First, express 32 as a power of 2, which is \(2^5\).
Then, apply the fifth root to the equation to get \( \root[5]{2^5} \).
Use the property that the nth root of \(a^n\) is 'a'. So, \( \root[5]{2^5} = 2 \).

These steps lead to a simplified result, confirming that the fifth root of 32 is 2.
By understanding these simplification steps, solving similar problems becomes easier.

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91影视

\(\sqrt[5]{32}\)

Short Answer

Step by step solution

Identify the problem

Rewrite 32 as a power of 2

Apply the fifth root

Simplify the expression

Conclusion

Key Concepts

Radicals

Exponents

Simplification

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