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Chapter 7: Problem 69

Simplify. Use absolute-value notation when necessary. $$ \sqrt[10]{(-6)^{10}} $$

Short Answer

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Step by step solution

01

- Understand the problem

The given expression is \( \sqrt[10]{(-6)^{10}} \). We need to simplify this expression and use absolute-value notation if necessary.
02

- Apply the Radical and Exponent Rule

Recall that \( \sqrt[n]{a^n} = |a| \) when \( n \) is even. Since \( 10 \) is even, we apply this rule. Here, \( a = -6 \) and \( n = 10 \). Therefore, \( \sqrt[10]{(-6)^{10}} = |-6| \).
03

- Evaluate the Absolute Value

The absolute value of \( -6 \) is \( 6 \), so \( |-6| = 6 \).
04

- Simplify the Expression

Combine the results of the previous steps. Therefore, the simplified form of \( \sqrt[10]{(-6)^{10}} \) is \( 6 \).

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Key Concepts

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

absolute value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number. For example, the absolute value of both -6 and 6 is 6. In mathematical notation, it is denoted as \(|a|\). The absolute value ensures that negative values become positive:
\(|-6| = 6\) and \(6 = 6\).

This concept is essential when you simplify expressions where even exponents are involved because raising a number to an even power always results in a non-negative value. Thus, for any number raised to an even power, its radical expression will involve absolute value to ensure the result is positive.
radical and exponent rules
Radicals and exponents are closely related. The general rule for radicals is:
\[ \sqrt[n]{a^n} = |a| \]

when \( n \) is even. This means that if you have an even root (like a square root, fourth root, etc.), the result must be non-negative. Let's break it down further:
  • The \( n \)-th root of a number \( a^n \) is the absolute value of \( a \).
  • This rule applies when the exponent is even () because an even exponent makes any negative base positive.
  • For odd exponents, the absolute value is not needed because the result can be negative or positive depending on the base.
Using our example \(\sqrt[10]{(-6)^{10}}\), we identify that 10 is an even exponent, requiring us to consider the absolute value of the base \(-6\):
\[ \sqrt[10]{(-6)^{10}} = |-6| \].
even exponents
An even exponent occurs when a number is raised to the power of an even number, such as 2, 4, 6, 8, etc. When you raise any real number to an even power, the result is always non-negative. Here are key points about even exponents:
  • Raising a positive number to an even exponent stays positive. For example, \((2^4 = 16)\).
  • Raising a negative number to an even exponent gives a positive result. For example, \((-2)^4 = 16)\).
The rule follows from the fact that multiplying an even number of negative factors results in a positive product, and multiplying an even number of positive factors remains positive. This is why in the context of radicals with even indices, the result necessitates the use of absolute value notation as we saw in \(\sqrt[10]{(-6)^{10}} = |-6|\).

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Most popular questions from this chapter

Contracting. Oxford Builders has an extension cord on their generator that permits them to work, with electricity, anywhere in a circular area of \(3850 \mathrm{ft}^{2} .\) Find the dimensions of the largest square room on which they could work without having to relocate the generator to reach each corner of the floor.

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