Problem 3 Change the following negative ex... [FREE SOLUTION]

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Painless Algebra

Lynette Long

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 7: Problem 3

Change the following negative exponents to positive exponents. $$ 2^{-5} $$

Short Answer

Expert verified

2^{-5} = \frac{1}{32}

Step by step solution

Understand Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, for any non-zero number 'a' and a positive integer 'n', the expression $a^{-n}$ is equivalent to $\frac{1}{a^n}$.

Apply the Negative Exponent Rule

Apply the rule $a^{-n} = \frac{1}{a^n}$ to the given problem. Here, the base is 2 and the exponent is -5. So, $2^{-5}$ can be rewritten as $\frac{1}{2^5}$.

Simplify the Expression

Now simplify the expression. Calculate $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$. Therefore, $\frac{1}{2^5} = \frac{1}{32}$.

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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 the reciprocal of a number, we mean flipping the number's fraction. Imagine you have a number 'a', the reciprocal is simply $\frac{1}{a}$. For example, the reciprocal of 2 is $\frac{1}{2}$. If you have a whole number, put it over 1 to turn it into a fraction, and then flip it. When dealing with negative exponents, you'll often use reciprocals. The rule is that $a^{-n}= \frac{1}{a^n}$. So in the case of our exercise, the reciprocal of $2^{-5}= \frac{1}{2^5}$. This is the foundation for simplifying expressions involving negative exponents.

Simplifying Expressions

Simplifying expressions involves breaking down more complex math problems into easier steps. When working with exponents, one helpful trick is applying the rules of exponents. Start by rewriting the expression using the exponent rules. In our exercise, we started with $2^{-5} $, which we rewrote as $ \frac{1}{2^5}$. Now, we simplify by calculating the exponent in the denominator. This means finding $2^5$, which equals 32. Therefore, $2^{-5}= \frac{1}{32}$. By simplifying step by step, you'll make complex expressions easier to manage.

Exponent Rules

Understanding exponent rules can dramatically simplify your math work. Some key exponent rules include:

$a^m \times a^n = a^{m+n} $
$a^m / a^n = a^{m-n} $
$ (a^m)^n = a^{m \times n} $
Negative exponents: $a^{-n} = \frac{1}{a^n} $

Our focus example uses the negative exponent rule: $2^{-5} = \frac{1}{2^5} $. We convert the negative exponent to a positive one by taking the reciprocal of the base. Then, we simplify the result of the exponentiation in the denominator to get $ \frac{1}{32} $. By mastering these rules, you can handle any problem involving exponents with ease.

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91影视

Change the following negative exponents to positive exponents. $$ 2^{-5} $$

Short Answer

Step by step solution

Understand Negative Exponents

Apply the Negative Exponent Rule

Simplify the Expression

Key Concepts

Reciprocal

Simplifying Expressions

Exponent Rules

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