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Chapter 10: Problem 17

Evaluate each exponential. $$ 16^{1 / 4} $$

Short Answer

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Step by step solution

01

Understand the Problem

The problem requires evaluating the exponential expression \(16^{1/4}\). This means finding the fourth root of 16.
02

Simplify the Exponent

Recall that an exponent of \(1/n\) means the nth root. So, \(16^{1/4}\) means the fourth root of 16.
03

Identify the Fourth Root

Determine which number, when raised to the power of 4, equals 16. \(2^4 = 16\), so the fourth root of 16 is 2.
04

Write the Answer

The value of \(16^{1/4}\) is 2.

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Key Concepts

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

Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, in the expression \(2^3\), the number 2 is the base, and 3 is the exponent. This means you multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).
Exponents can also be fractions. These fractional exponents are closely related to roots, such as square roots or cube roots. The numerator of the fraction indicates the power, and the denominator indicates the root. For instance, \(16^{1/4}\) means the fourth root of 16.
Understanding how exponents work is crucial for manipulating and simplifying expressions involving them.
Exploring Roots and Radicals
Roots and radicals are used to undo the action of exponents. For example, the square root of 9 is 3, because \(3^2 = 9\). Radicals can be expressed using a different notation, involving the radical symbol (鈭�). The square root of a number \(x\) is written as \(\sqrt{x}\).
Higher-order roots like the cube root and fourth root can be written similarly. For example, the fourth root of 81 is written as \(\sqrt[4]{81}\). In terms of exponents, this is the same as \(81^{1/4}\).Knowing how to handle these roots is important because it allows for the simplification and evaluation of more complex expressions.
Simplifying Expressions
Simplifying expressions is about making them easier to understand or solve. This often involves combining like terms, factoring, or using exponent rules. When working with exponents and roots, simplification might include converting between radical and exponential forms.
For example, consider the expression \(16^{1/4}\). The exponent 1/4 tells us to find the fourth root of 16. We know that \(2^4 = 16\), so the fourth root of 16 is 2. Therefore, \(16^{1/4} = 2\).
Simplifying expressions correctly helps you get to the right answer more efficiently and makes it easier to understand the underlying mathematical concepts.

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