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Chapter 7: Problem 70

Simplify. Use absolute-value notation when necessary. $$ \sqrt[12]{(-10)^{12}} $$

Short Answer

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Step by step solution

01

Identify the Problem

The task is to simplify the expression \( \sqrt[12]{(-10)^{12}} \ \).
02

Rewrite the Expression Using Properties of Exponents

Recognize that \( \sqrt[12]{a^{12}} = |a| \ \), where \( a \ \) is any real number. This property of exponents will help in simplifying the given expression.
03

Apply the Rule

Using the rule, rewrite \( \sqrt[12]{(-10)^{12}} \ \) as \( |-10| \ \).
04

Simplify the Absolute Value

The absolute value of \( -10 \ \) is \( 10 \ \). Therefore, \( |-10| = 10 \ \).

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Key Concepts

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

Absolute Value
Let's start by understanding the **absolute value**. The absolute value of a number is its distance from zero on the number line, regardless of direction. For instance, both +5 and -5 have an absolute value of 5.
This means that absolute value makes numbers positive. In symbols, we write the absolute value of a number 'a' as |a|. Here are some examples:
  • |7| = 7
  • |-3| = 3

In the problem, we found that the absolute value of -10 is 10, so |-10| = 10.
Properties of Exponents
Now, let's dive into the **properties of exponents**. Exponents tell us how many times to multiply a base number by itself. For example, in the expression \(5^3\), 5 is the base, and 3 is the exponent, representing \(5 \times 5 \times 5\).
Here are some critical properties you should know:
  • Product of Powers: \( a^m \times a^n = a^{m+n} \)
  • Power of a Power: \( (a^m)^n = a^{m \times n} \)
  • Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)

In our problem, we used the property \( \sqrt[n]{a^n} = |a| \). This property tells us that if we take an nth root of a number raised to the nth power, the result is the absolute value of the base.
Radicals
Lastly, let's explore **radicals**. A radical expression involves roots, such as square roots or cube roots. The general form of a radical is \( \sqrt[n]{a} \), where 'n' is the root index and 'a' is the radicand.
Here are some key points about radicals:
  • The square root of a number 'a' is written as \( \sqrt{a} \) and indicates a number whose square is 'a'.
  • The cube root of a number 'a' is written as \( \sqrt[3]{a} \) and indicates a number whose cube is 'a'.

In the problem we had \( \sqrt[12]{(-10)^{12}} \), which simplifies using the principles we discussed.
By recognizing that taking the 12th root of \( (-10)^{12} \) simplifies to the absolute value, we didn't need to perform each calculation step-by-step. Thus, we directly found \( |-10| = 10 \).

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Most popular questions from this chapter

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt[3]{a} \sqrt[6]{a} $$

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