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Chapter 7: Problem 68

Simplify. Use absolute-value notation when necessary. $$ \sqrt[4]{(7 b)^{4}} $$

Short Answer

Expert verified
The simplified form is \( |7b| \).

Step by step solution

01

- Understanding the Expression

The expression given is \( \sqrt[4]{(7b)^{4}} \). This is a fourth root of a term raised to the fourth power.
02

- Simplify the Root and Power

Since \sqrt[n]{a^{n}} = |a|\, we can simplify \( \sqrt[4]{(7b)^{4}} \) to \( |7b| \).
03

- Write the Final Answer

The final simplified form of the expression is \( |7b| \).

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Key Concepts

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

absolute value
In mathematics, the absolute value of a number refers to its distance from zero on a number line. This value is always non-negative. For example, the absolute value of both 3 and -3 is 3. To represent the absolute value of a number, we use two vertical bars, like this: \(|x|\).

When simplifying expressions that involve roots and powers, absolute value notation is sometimes necessary to ensure the result is always non-negative. For instance, the simplification of \(\sqrt[4]{(7b)^4}\) relies on understanding absolute values because we want the expression to stay positive, which leads us to \(|7b|\).
  • If the original term is positive or zero, the absolute value is straightforward: just the term itself.
  • If the original term is negative, the absolute value negates it, making it positive.
fourth root
The fourth root of a number is one of the numbers that, when raised to the fourth power, gives the original number. For instance, the fourth root of 16 is 2 because \(2^4 = 16\).

The notation for the fourth root is \(\sqrt[4]{x}\). In our exercise, we have \(\sqrt[4]{(7b)^4}\). The fourth root and the fourth power are inverse operations, meaning they cancel each other out. However, due to properties of roots and powers in mathematics, when we simplify, we must ensure the result is always non-negative, which is why we use absolute value notation.
  • For example, \(\sqrt[4]{81} = 3\) because \(3^4 = 81\).
  • Similarly, \(\sqrt[4]{16} = 2\) because \(2^4 = 16\).
powers and roots
Understanding powers and roots is crucial in algebra. A power indicates how many times to multiply a number by itself. For example, \(7^2\) means multiplying 7 by itself, which equals 49. To express this mathematically: \(a^n\) means multiplying \(a\) by itself \(n\) times.

Conversely, a root denotes the inverse operation of a power. The most common roots are square roots \(\sqrt{x}\) and fourth roots \(\sqrt[4]{x}\). Simplifying powers and roots involves ensuring the expression is in its simplest form. In our problem \(\sqrt[4]{(7b)^4}\), taking the fourth root of \((7b)^4\) simplifies directly to \(|7b|\).

This relationship relies on key properties:
  • \(\sqrt[n]{a^n} = |a|\)
  • \((a^m)^n = a^{m*n}\)
  • Both operations must follow these rules to maintain certainty and correctness.

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Most popular questions from this chapter

Simplify. $$(5 i)^{3}$$

Find a simplified form for \(f(x) .\) Assume \(x \geq 0\) $$ f(x)=\sqrt[4]{x^{5}-x^{4}}+3 \sqrt[4]{x^{9}-x^{8}} $$

Classify each of the following as either an arithmetic sequence, a geometric sequence, an arithmetic series, a geometric series, or none of these. \(3,6,12,24, \dots\)

Simplify. $$(1-i)^{3}(1+i)^{3}$$

Find the midpoint of the segment with the given endpoints. $$ (2,-1) \text { and }(5,8) $$
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