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Chapter 4: Problem 4

Evaluate each expression. Do not use a calculator. $$\sqrt[4]{16}$$

Short Answer

Expert verified
The fourth root of 16 is 2.

Step by step solution

01

Identify the Expression

The expression to evaluate is \( \sqrt[4]{16} \). This means we need to find the fourth root of 16.
02

Understand the Fourth Root

The fourth root of a number is a value that, when raised to the power of 4, equals the original number. Here, we are looking for a number \( x \) such that \( x^4 = 16 \).
03

Find the Fourth Root

Consider potential whole numbers that could satisfy the equation \( x^4 = 16 \). Test small numbers: - \( 1^4 = 1 \), too small.- \( 2^4 = 16 \), which matches our requirement. So, \( x = 2 \).
04

Conclusion

After testing, we find that \( 2^4 = 16 \), meaning the fourth root of 16 is 2. Thus, \( \sqrt[4]{16} = 2 \).

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Key Concepts

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

Fourth Root
The fourth root of a number is a unique mathematical concept that helps us find a specific value. This value, when raised to the power of four, equals the original number. For example, with \( \sqrt[4]{16} \), we want to determine a number \( x \) such that \( x^4 = 16 \). To find this value, you don't need a calculator. You just need to test small numbers manually.
  • Start with small numbers and calculate their powers of four.
  • For instance, \( 1^4 = 1 \) and \( 2^4 = 16 \).
You can easily see that when \( 2 \) is raised to the fourth power, it matches 16 exactly. Thus, the fourth root of 16 is 2.
Exponents
Exponents are a way to express repeated multiplication of the same number. In the expression \( x^n \), \( x \) is the base, and \( n \) is the exponent, indicating that \( x \) is multiplied by itself \( n \) times.In the context of our example, when we say \( 2^4 \), we mean \( 2 \times 2 \times 2 \times 2 \). This method simplifies how we represent larger calculations compactly. Calculating with exponents can make solving expressions such as finding roots faster.Key tips when dealing with exponents:
  • The base is the number being multiplied.
  • The exponent shows how many times to multiply the base by itself.
  • Understanding basic exponents can pave the way for tackling more complex mathematical problems.
Evaluating Expressions
Evaluating expressions involves applying operations to determine the value of the expression. One must carefully follow mathematical rules and processes to reach the correct answer.Here's how to efficiently evaluate expressions:
  • Start by identifying the type of mathematical operation you're dealing with, such as addition, subtraction, multiplication, or roots.
  • Apply the appropriate rules or methods involved in the operation. For roots, consider numbers' properties and apply exponentiation principles.
  • Always verify your final answer by plugging it back into the original expression to ensure accuracy.
Evaluating \( \sqrt[4]{16} \) requires finding what number to raise to the power of four to get 16. By following these steps, you establish that \( 2 \) is the answer, as \( 2^4 = 16 \). This confirms your solution and completes the expression evaluation process.

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Most popular questions from this chapter

Solve the equation in part (a) graphically, expressing solutions to the nearest hundredth. Then, use the graph to solve the associated inequalities in parts (b) and (c), expressing endpoints to the nearest hundredth. (a) \(\frac{\sqrt[3]{7} x^{3}-1}{x^{2}+2}=0\) (b) \(\frac{\sqrt[3]{7} x^{3}-1}{x^{2}+2}>0\) (c) \(\frac{\sqrt[3]{7} x^{3}-1}{x^{2}+2}<0\)

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