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Chapter 9: Problem 68

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ 8^{-1 / 3} $$

Short Answer

Expert verified
The expression simplifies to \(\frac{1}{2}\).

Step by step solution

01

Understand the Meaning of the Expression

We are given the expression \(8^{-1/3}\). This is represented as a power expression where 8 is the base, and \(-1/3\) is the exponent.
02

Apply Negative Exponent Rule

A negative exponent indicates that we can take the reciprocal of the base. Thus, \(8^{-1/3} = \frac{1}{8^{1/3}}\). Now, we need to focus on calculating \(8^{1/3}\).
03

Evaluate the Cube Root

The expression \(8^{1/3}\) implies finding the cube root of 8. Since \(2^3 = 8\), \(8^{1/3} = 2\).
04

Simplify the Expression

Replace \(8^{1/3}\) with 2 in the reciprocal expression. Thus, \(\frac{1}{8^{1/3}} = \frac{1}{2}\).
05

Final Answer

The simplified expression of \(8^{-1/3}\) is \(\frac{1}{2}\).

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Key Concepts

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

Negative Exponents
In mathematics, a negative exponent is a way to signify the reciprocal of a positive exponent. When you see an expression like \(8^{-1/3}\), the negative exponent means that you first take the reciprocal of the base number raised to the positive version of that exponent.

Here's a step-by-step look at how to handle negative exponents:
  • Identify the base and the negative exponent. For example, in \(8^{-1/3}\), 8 is the base, and \(-1/3\) is the exponent.
  • Convert the expression to its reciprocal by changing the negative exponent into a positive. Use the formula: \(a^{-m} = \frac{1}{a^m}\). So, \(8^{-1/3}\) becomes \(\frac{1}{8^{1/3}}\).
This conversion simplifies the problem, allowing you to focus on solving the expression with a positive exponent.
Cube Roots
Cube roots are the opposite operation of cubing a number. To cube a number means to multiply it by itself twice, whereas a cube root aims to find a number that, when cubed, gives the original number. For example, when dealing with \(8^{1/3}\):

To find the cube root of a number:
  • Identify the number you need the cube root of, which is the base in our expression \(8^{1/3}\).
  • Think of the number that when multiplied by itself three times yields the base number. So, since \(2 \times 2 \times 2 = 8\), \(2^3 = 8\), this means the cube root of 8 is 2.
Cube roots are helpful in simplifying expressions involving fractional exponents. Recognizing this relationship speeds up calculation and simplification.
Reciprocals
Understanding reciprocals is crucial when working with expressions involving inverse operations. A reciprocal of a number is essentially 1 divided by that number. For example, the reciprocal of 2 is \(\frac{1}{2}\).

When you encounter a negative exponent like in \(8^{-1/3}\), the concept of reciprocals helps transform the expression:
  • The negative exponent \(-1/3\) initially suggests that 8 should be handled as a reciprocal, resulting in \(\frac{1}{8^{1/3}}\).
  • Once you evaluate the cube root of 8 to find it equals 2, you replace and simplify the expression to \(\frac{1}{2}\).
Reciprocals simplify expressions and allow us to handle negative exponents intuitively by flipping the fractions.

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Most popular questions from this chapter

If \(x\) is any real number, that is, if \(x\) is unrestricted, then \(\sqrt{x^{2}}=x\) is not correct. Explain why.

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[3]{-216 z^{9}} $$

Simplify each expression. All variables represent positive real numbers. $$ \left(16 x^{4}\right)^{1 / 4} $$

Use a calculator to evaluate each expression. Round to the nearest hundredth. See Using Your Calculator: Rational Exponents. $$ (50.5)^{1 / 4} $$

Simplify each expression. All variables represent positive real numbers. a. \(81^{1 / 4}\) b. \(81^{-1 / 4}\) c. \(-81^{1 / 4}\) d. \(\frac{1}{81^{-1 / 4}}\)
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