Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$32^{1 / 5}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Radical Expression

The expression \(32^{1/5}\) is a radical expression. When you see a fractional exponent, the numerator indicates the power, and the denominator indicates the root. In this case, \(32^{1/5}\) means we need to find the 5th root of 32.

Step 2: Express as a Root

Convert the expression \(32^{1/5}\) into radical form. This can be written as \(\sqrt[5]{32}\), which means we are looking for a number that, when raised to the power of 5, equals 32.

Step 3: Calculate the 5th Root of 32

To find \(\sqrt[5]{32}\), try to estimate which integer raised to the fifth power equals 32. Check small whole numbers, starting from 1, and test until you find: \(2^5 = 32\). This confirms that the 5th root of 32 is 2.

Key Concepts

Fractional Exponents

Fractional exponents can initially be a bit intimidating, but once you understand what they signify, they become manageable tools in algebra. A fractional exponent like \(a^{m/n}\) tells you two operations: the root and the power.

• The numerator \(m\) indicates the power to which the base \(a\) is raised.
• The denominator \(n\) tells you to find the \(n^{th}\) root of the base.

This means \(a^{m/n}\) can be written in radical form as \((\sqrt[n]{a})^m\). For instance, \(32^{1/5}\) indicates the 5th root of 32 because the fraction is \(1/5\). Here, "1" is the power and "5" is the root. Thus, the expression can also be interpreted as first finding the 5th root of 32, and then raising the result to the power of 1, which essentially keeps the number the same.

Roots

Roots are essential in simplifying algebraic expressions, especially with fractional exponents. When you are asked to calculate \(\sqrt[n]{a}\), it means you need to find a number that, when multiplied by itself "n" times, gives \(a\). This is called the \(n^{th}\) root.
In our example, \(\sqrt[5]{32}\) tells us to find a number \(x\) such that \(x^5 = 32\).
Roots come in various kinds:

• Square Roots: These have an index of 2 (often just written as \(\sqrt{a}\)), such as \(\sqrt{9} = 3\).
• Cubic Roots: These are the 3rd roots of a number, like \(\sqrt[3]{27} = 3\).

In the specific context of \(32^{1/5}\), we find that \(2\) raised to the 5th power produces 32, hence \(\sqrt[5]{32} = 2\). Understanding how to find roots is crucial for mastering radical expressions.

Integer Powers

Understanding integer powers is a key part of working with radical expressions and simplifies many mathematical operations. An integer power \(a^n\) involves multiplying the base \(a\) by itself \(n\) times.
For example, the expression \(2^5\) means you multiply 2 by itself 5 times, which results in \(32\). Breaking down, you do the following multiplications:

• \(2 \times 2 = 4\)
• \(4 \times 2 = 8\)
• \(8 \times 2 = 16\)
• \(16 \times 2 = 32\)

In our work with \(32^{1/5}\), recognizing that \(32 = 2^5\) helps us see that the 5th root of 32 is simply 2. This reveals the beauty and power of integer exponents, which simplify many aspects of working with numerical expressions. As you gain familiarity, these calculations become second nature.