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Chapter 10: Problem 98

\(\sqrt[4]{16}\)

Short Answer

Expert verified
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Step by step solution

01

Understand the Exercise

The task is to find the fourth root of 16, written as \(\sqrt[4]{16}\). This means determining a number that, when multiplied by itself four times, equals 16.
02

Rewrite 16 as a Power of 2

Recognize that 16 can be written as a power of 2: \(16 = 2^4\).
03

Apply the Fourth Root

Take the fourth root of both sides of the equation. Since we have \(\sqrt[4]{2^4} = 2\), 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.

importance of understanding roots of numbers
Roots of numbers are essential in mathematics because they help us find a number that, when multiplied by itself a certain number of times, gives another number. For instance, the fourth root of 16 means finding a number that, when multiplied by itself four times, equals 16. Knowing how to calculate roots helps in solving more complex algebraic expressions and equations.
  • Square roots involve multiplying a number by itself once.
  • Cube roots involve multiplying a number by itself twice.
  • Fourth roots involve multiplying a number by itself three times.
This foundational knowledge is crucial for mastering higher-level math topics.
spotting exponents in algebraic expressions
Understanding exponents is fundamental to grasping algebraic expressions. Exponents indicate how many times a number, known as the base, is multiplied by itself. For example, in the exercise, 16 can be written as a power of 2: 16 = 2^4. This tells us that 2 is the base and 4 is the exponent, meaning 2 is multiplied by itself four times to get 16.
  • Exponents simplify repeated multiplication.
  • They are written as a small number to the upper-right of the base number.
  • It鈥檚 essential to recognize when numbers can be expressed in exponential form to simplify calculations.
Grasping exponents helps in effectively breaking down and solving algebraic problems.
working with algebraic expressions
Algebraic expressions involve numbers, variables, and operation signs. Understanding how to manipulate these expressions is key to solving algebra problems. For example, rewriting 16 as an exponential term, 16 = 2^4, helps in determining the fourth root. This process simplifies the calculation and makes it easier to solve.
  • Algebraic expressions can include constants, variables, coefficients, and exponents.
  • Knowing how to rewrite expressions in simpler forms is crucial.
  • This skill aids in breaking down complex problems into manageable parts.
Mastering algebraic expressions is essential for advancing in mathematics and solving equations efficiently.

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Most popular questions from this chapter

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{300 z^{3}}\)

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{72 k^{2}}\)

List all of the following sets to which each number belongs. A number may belong to more than one set. real numbers pure imaginary numbers nonreal complex numbers complex numbers $$ \sqrt{2} $$

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$ \sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b $$ Explain why this is faulty reasoning.

Simplify. Assume that all variables represent positive real numbers. \(\sqrt[3]{y} \cdot \sqrt[4]{y}\)
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