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Chapter 4: Problem 16

Evaluate each expression. Do not use a calculator. $$32^{1 / 5}$$

Short Answer

Expert verified
The value of \(32^{1/5}\) is 2.

Step by step solution

01

Understanding the Problem

We want to evaluate the expression \(32^{1/5}\). This notation means we are looking for the fifth root of 32, which is the number that, when multiplied by itself five times, yields 32.
02

Break Down into Prime Factors

Start by breaking down 32 into its prime factors. Note that 32 is a power of 2, so \(32 = 2 imes 2 imes 2 imes 2 imes 2 = 2^5\).
03

Apply the Fractional Exponent

Since \(32 = 2^5\), we can rewrite \(32^{1/5}\) using the property of exponents: \( (2^5)^{1/5} = 2^{(5 imes 1/5)} = 2^1\).
04

Simplify the Expression

Simplify \(2^1\) to get 2. This step confirms that the original expression \(32^{1/5}\) simplifies to 2.

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Key Concepts

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

Fifth Root
The fifth root of a number is the value that, when multiplied by itself five times, equals that number.
Finding the fifth root is like asking, "What number do I multiply by itself five times to get back to the original number?"
For the number 32, the fifth root is 2, because \(2 \times 2 \times 2 \times 2 \times 2 = 32\).
Fifth roots are simply an extension of the more familiar square and cube roots. You can also represent the fifth root using fractional exponents, where the exponent is \(1/5\).
This idea brings us to the concept of fractional exponents, which we will explore next.
Prime Factorization
Prime factorization is the process of breaking down a number into its most basic building blocks 鈥� the prime numbers.
A prime number is a number greater than 1 that has no other divisors except for 1 and itself.
The prime factorization of 32 is an important step in simplifying expressions like \(32^{1/5}\).
  • Start with the number 32.
  • Divide by the smallest prime number (2) and continue dividing until you cannot go further.
You will find that \(32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5\).
This representation is useful when applying rules of exponents, as it transforms complex numbers into easily manageably forms.
Fractional Exponents
Fractional exponents are a way of expressing roots using exponents.
They allow us to represent roots and powers as a single operation. For example, the fifth root of a number can be written as an exponent of \(1/5\).
  • This representation stems from the basic property of exponents: \( (a^m)^n = a^{m \times n} \).
  • When you see \(32^{1/5}\), you can think of it as \((32)^1\) raised to the power of \(1/5\).
The benefit of this approach is that it simplifies complex operations into straightforward calculations.
By converting roots into fractional exponents, you can easily handle more complex mathematical equations and calculations.

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Most popular questions from this chapter

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