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Chapter 8: Problem 59

Find the indicated root, or state that the expression is not a real number. $$\sqrt[4]{16}$$

Short Answer

Expert verified
The fourth root of 16 is \(2\).

Step by step solution

01

Understanding the Term

The expression \(\sqrt[4]{16}\) is asking for the fourth root of 16, which is the number that if multiplied by itself four times equals 16.
02

Calculating the Root

Consider the numbers 1 through 5 and examine their fourth powers (1^4, 2^4, etc.) to see which one equals 16. It鈥檚 found that 2^4 = 16.
03

Asserting the Answer

After calculating and comparing, it becomes clear that 2 is the number that when raised to the power of 4 equals 16. As a result, we can deduce that the fourth root of 16 is 2.

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Key Concepts

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

Understanding the Fourth Root
The concept of a fourth root might seem complex at first. However, it becomes much simpler when we break it down.
  • In mathematics, the fourth root of a number is a value that, when multiplied by itself four times, gives the original number.
  • This can be written using radical notation as \(\sqrt[4]{x}\), where \(x\) is the number whose fourth root is being calculated.
For example, in \(\sqrt[4]{16}\), we are looking for a number that gives 16 when multiplied by itself four times. This number is 2, because \(2 \times 2 \times 2 \times 2 = 16\). Knowing how to find the fourth root is a useful skill. It helps in understanding more complex mathematical ideas later on.
Decoding Exponents
Exponents are a key part of algebra that help us represent repeated multiplication in a compact form.
  • When we talk about exponents, we usually see expressions like \(x^n\). Here, \(x\) is the base, and \(n\) is the exponent.
  • Exponents tell us how many times to multiply the base by itself.
Thus for \(2^4\), 2 is the base and 4 is the exponent, meaning you multiply 2 by itself 4 times: \(2 \times 2 \times 2 \times 2\). This equals 16, showing the connection between exponents and finding roots, such as the fourth root. Understanding exponents is essential in algebra. They pave the way for solving complex equations and understanding higher-level math concepts.
Exploring Real Numbers
Real numbers are a broad category of numbers that are used to measure quantities and describe linguistics in mathematics.
  • They include all the numbers that you are likely familiar with, such as positive numbers, negative numbers, and zero.
  • Real numbers can be divided into rational numbers (like fractions and integers) and irrational numbers (which cannot be expressed as a simple fraction, like \(\pi\) or square root of 2).
When we calculate roots, such as \(\sqrt[4]{16}\), we are determining if the result is a real number. In this case, the fourth root of 16 is 2, which is a real number. Understanding real numbers helps us make sense of the world around us mathematically, as they provide a complete set of numbers for performing algebraic operations.

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Most popular questions from this chapter

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$\left(\frac{9}{100}\right)^{-\frac{1}{2}}$$

Rationalize the denominator: \(\frac{1}{\sqrt[3]{2}}\)

Without using a calculator, simplify the expressions completely. $$25^{\frac{1}{4}} \cdot 25^{-\frac{3}{4}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The first step in solving \(\sqrt{x}+3=4\) is to square each side.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no question that \((-64)^{\frac{1}{3}}=-64^{\frac{1}{3}},\) so I can conclude that \((-64)^{\frac{1}{2}}=-64^{\frac{1}{2}}\).
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