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Chapter 10: Problem 40

Rewrite with a positive exponent and evaluate. \(-\left(\frac{1}{125}\right)^{-1 / 3}\)

Short Answer

Expert verified
Rewriting the expression with a positive exponent and evaluating it, we get: \(-\left(\frac{1}{125}\right)^{-\frac{1}{3}} = -\frac{1}{\left(\frac{1}{125}\right)^{\frac{1}{3}}} = -\frac{1}{\frac{1}{5}} = -5\)

Step by step solution

01

Remove the negative exponent

We can use the property of negative exponents to rewrite the expression. In our case, \(-\left(\frac{1}{125}\right)^{-\frac{1}{3}} = -\frac{1}{\left(\frac{1}{125}\right)^{\frac{1}{3}}}\)
02

Evaluate the expression with a positive exponent

Now we need to find the cube root of \(\frac{1}{125}\). Since \(125 = 5^3\), we get \(\left(\frac{1}{125}\right)^{\frac{1}{3}} = \frac{1}{5}\)
03

Substitute the positive exponent result back into the expression

Now substitute the result from step 2 back into the expression from step 1: \(-\frac{1}{\left(\frac{1}{125}\right)^{\frac{1}{3}}} = -\frac{1}{\frac{1}{5}}\)
04

Evaluate the final expression

To evaluate the expression \(-\frac{1}{\frac{1}{5}}\), we can find a reciprocal of the fraction in the denominator: \(-\frac{1}{\frac{1}{5}} = -1 \times \frac{5}{1} = -5\) So the final result is: \(-\left(\frac{1}{125}\right)^{-\frac{1}{3}} = -5\)

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Key Concepts

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

Evaluating Expressions
Evaluating expressions is all about determining the value of a mathematical statement. You begin by following the operations outlined in the expression. Look at each term closely and apply mathematical rules in sequence.
To better understand:
  • Identify the operations involved, such as addition, subtraction, multiplication, division, etc.
  • Follow the **order of operations**, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
  • Simplify each part of the expression carefully.
In the given exercise, the expression \(-\left(\frac{1}{125}\right)^{-1 / 3}\) involves a negative exponent. Here, we must use the rule of negative exponents first. Once rewritten with a positive exponent, subsequent steps can follow using known mathematical principles.
Cube Roots
Cube roots are related to the operation of factoring numbers into their cubic components. Essentially, finding the cube root involves determining which number, when multiplied by itself three times, results in the original number.
For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
  • To find the cube root of a fraction like \(\frac{1}{125}\), factor the number in the denominator.
  • In this exercise, 125 can be expressed as \(5^3\).
  • Thus, the cube root of \(\frac{1}{125}\) is \(\frac{1}{5}\).
This understanding allows you to simplify mathematical expressions where cube roots are involved, making the overall evaluation easier.
Properties of Exponents
Exponents have specific properties that simplify the manipulation and evaluation of expressions. These properties are essential for handling both positive and negative exponents.
Consider the following key properties:
  • **Negative Exponent Rule**: \(a^{-n} = \frac{1}{a^n}\), which turns the base into its reciprocal and changes the sign of the exponent to positive.
  • **Exponent of a Fraction Rule**: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), which applies the exponent to both the numerator and the denominator.
  • **Power of a Power Rule**: \((a^m)^n = a^{m \times n}\), which simplifies when a power is raised to another power.
In solving expressions with exponents, recognizing and applying these properties correctly can greatly simplify the process and lead to accurate solutions. In this exercise, the negative exponent is transformed using the Negative Exponent Rule, making the problem more straightforward to solve.

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