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Chapter 10: Problem 37

Write with positive exponents. Simplify if possible. $$ \frac{1}{a^{-2 / 3}} $$

Short Answer

Expert verified
Rewriting the expression, the answer is \( a^{2/3} \).

Step by step solution

01

Understand Negative Exponents

The first step requires understanding what a negative exponent signifies. Recall that a negative exponent, such as in the expression \( a^{-n} \), represents the reciprocal of the base raised to the positive exponent: \( a^{-n} = \frac{1}{a^n} \). Hence, \( a^{-2/3} = \frac{1}{a^{2/3}} \).
02

Simplify the Expression

In the original expression, \( \frac{1}{a^{-2/3}} \), the base is already in the denominator. Rewriting it using positive exponents, we get \( \frac{1}{\frac{1}{a^{2/3}}} \).
03

Solve for Positive Exponent

Since we have a fraction within a fraction, we can simplify \( \frac{1}{\frac{1}{a^{2/3}}} \) by multiplying by the reciprocal. This turns into \( a^{2/3} \).
04

Final Expression with Positive Exponents

Thus, the expression \( \frac{1}{a^{-2/3}} \) becomes \( a^{2/3} \) when rewritten with positive exponents.

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Key Concepts

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

Reciprocal
When we talk about reciprocals, it's essential to understand what it means in the context of exponents and fractions.
The reciprocal of a number or expression is essentially what you multiply by to get 1. In simpler terms, if you have a number like 5, its reciprocal is \( \frac{1}{5} \). It's similar when dealing with exponents.
  • If you have a base with a negative exponent, such as \( a^{-n} \), it signifies the reciprocal of the base raised to the corresponding positive exponent.
  • So, \( a^{-n} = \frac{1}{a^n} \).
In use, this concept allows negative exponents to be rewritten in a more manageable form that involves positive exponents, leading us to the next topic.
Positive Exponents
Once we've understood reciprocals, we can now focus on turning negative exponents into positive ones.
Positive exponents signify standard repeated multiplication. A positive exponent, \( a^n \), means multiplying the base \( a \) by itself \( n \) times.
  • A negative exponent flips the base to its reciprocal, hence turning a negative exponent into a positive exponent is a step that simplifies calculations.
  • For example, converting a negative exponent like \( a^{-2/3} \) to a positive one would result in \( \frac{1}{a^{2/3}} \).
If we have an expression \( \frac{1}{a^{-x}} \), this can be simplified using reciprocals and positive exponents by reflecting the idea that \( \frac{1}{\frac{1}{a^x}} \) simplifies directly to \( a^x \).
Simplifying Expressions
Simplifying expressions is all about reducing them to their simplest form.
This is particularly helpful in mathematics to make expressions easier to interpret and solve.
  • Simplification involves using properties such as the reciprocal and positive exponents to rewrite an expression in a less complicated way.
  • In the context of the problem \( \frac{1}{a^{-2/3}} \), simplification involves understanding that the base with a negative exponent can be simplified to a positive exponent: \( a^{2/3} \).
By performing such simplifications, expressions become more intuitive and less cumbersome to work with, which is valuable in solving more complex problems efficiently.

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Most popular questions from this chapter

Simplify each exponential expression. $$\left(-14 a^{5} b c^{2}\right)\left(2 a b c^{4}\right)$$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$f(3)$$

Use a calculator to write a four-decimal-place approximation of each number. $$ 76^{5 / 7} $$

Identify the domain and then graph each function. $$f(x)=\sqrt{x}-2$$

Factor the common factor from the given expression. $$ x^{2 / 7} ; x^{3 / 7}-2 x^{2 / 7} $$
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