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Chapter 7: Problem 67

Simplify. Use absolute-value notation when necessary. $$ \sqrt[4]{(6 a)^{4}} $$

Short Answer

Expert verified
\(|6a|\)

Step by step solution

01

Understand the Problem

The problem requires simplifying the fourth root of \( (6a)^{4} \).
02

Rewrite the Expression

Rewrite the expression under the fourth root: \( \sqrt[4]{(6a)^{4}} \). This is asking for the fourth root of \( (6a)^{4} \).
03

Use the Property of Roots and Exponents

Recall that \( \sqrt[n]{x^{n}} = |x| \) when \( n \) is an even number. In this case, \( n = 4 \).
04

Apply the Property

Applying the property, we get \( \sqrt[4]{(6a)^{4}} = |6a| \). The absolute value notation is necessary because the fourth root of a fourth power must yield a non-negative result.
05

Simplify the Expression

Finally, the expression simplifies to \( |6a| \).

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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 special mathematical operation. It helps us determine what value, when multiplied by itself four times, would give the original number. For example, the fourth root of 16 is 2 since 2*2*2*2 equals 16.
To simplify an expression like \(\sqrt[4]{(6a)^4}\), we look for the number that, when raised to the power of 4, equals \((6a)^4\).
This can be written as \(\sqrt[4]{(6a)^4} = (|6a|)\), thanks to the properties of exponents and absolute value notation.
Absolute Value
Absolute value is the non-negative value of a number, regardless of its sign. For example, the absolute value of -5 is 5.
In mathematical terms, the absolute value of \(x\) is denoted as \(|x|\).
When dealing with roots and exponents, especially when finding even roots like the fourth root, the absolute value ensures that the result is always non-negative.
For the expression \(\sqrt[4]{(6a)^4}\), using absolute value notation is necessary to ensure the result is non-negative, resulting in \(|6a|\).
Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, \(3^4\) means \(3*3*3*3\).
In our problem \(\sqrt[4]{(6a)^4}\), we see an exponent inside the root.
The formula \(\sqrt[n]{\left(x\right)^n} = |x|\) helps us simplify such expressions, particularly when dealing with even roots and powers. Applying this property, \(\sqrt[4]{(6a)^4} = |6a|\), allows us to simplify the expression accurately.

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Most popular questions from this chapter

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