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Chapter 8: Problem 63

What is the meaning of \(a^{-\frac{m}{n}} ?\) Give an example.

Short Answer

Expert verified
The equation \( a^{-\frac{m}{n}} \) is equal to \( \sqrt[n]{\frac{1}{a^m}} \). An example, if \( a = 2, m = 2, n = 1 \), then \( a^{-\frac{m}{n}} = 0.5 \)

Step by step solution

01

Understand the exponent rules

Firstly, an exponent refers to the number of times a number is multiplied by itself. For instance, \( a^3 \) equals \( a \times a \times a \). For \( a^{-m} \), it implies that the number a is in the denominator, i.e., \( a^{-m} = \frac{1}{a^m} \).
02

Fractional exponents

A fractional exponent \( a^{\frac{1}{n}} \), denotes the n-th root of a. For instance, \( a^{\frac{1}{2}} \) is the square root of a. The base number \( a^{\frac{m}{n}} \) is equal to the n-th root of \( a^m \), denoted as \( \sqrt[n]{a^m} \).
03

Combine the two concepts

So if we have a negative fractional exponent like \( a^{-\frac{m}{n}} \), it means \( \frac{1}{a^{\frac{m}{n}}} \), or \( \frac{1}{\sqrt[n]{a^m}} \). This formula can be further simplified into \( \sqrt[n]{\frac{1}{a^m}} \).
04

Give an example

Let's assume \( a = 2, m = 2, n = 1 \). According to the formula we have \( a^{-\frac{m}{n}} = \sqrt[\frac{1}{n}]{\frac{1}{a^m}} = \sqrt{ \frac{1}{2^2}} = \sqrt{0.25} = 0.5 \)

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Most popular questions from this chapter

Exercises \(105-107\) will help you prepare for the material covered in the next section. $$\text { Solve: } \quad 7=-2.5 x+17$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$16^{-\frac{1}{4}}=-2$$

Simplify each expression. Write answers in exponential form with positive exponents only. Assume that all variables represent positive real numbers. $$\left(x^{\frac{1}{4}} y^{3}\right)^{\frac{2}{3}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$81^{\frac{1}{4}} \cdot 125^{\frac{1}{3}} \text { is an integer. }$$

Will help you prepare for the material covered in the first section of the next chapter. Use substitution to determine if \(-\sqrt{5}\) is a solution of the quadratic equation \(4 x^{2}=20\).
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