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Chapter 5: Problem 14

Express using positive exponents and, if possible, simplify. $$(-3)^{-4}$$

Short Answer

Expert verified
The simplified form is \(\frac{1}{81}\).

Step by step solution

01

Understand Negative Exponents

Recall that a negative exponent indicates that we take the reciprocal of the base and change the sign of the exponent to positive. That means \((-3)^{-4}\text{ can be written as }\frac{1}{(-3)^4}\).
02

Calculate the Positive Exponent

Compute \( (-3)^4 \). This is done by multiplying -3 by itself four times: \(-3 \times -3 \times -3 \times -3 = 81\).
03

Simplify the Result

Using the result from Step 2, \((-3)^{-4}\text{ is equal to }\frac{1}{81}\).

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Key Concepts

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

Positive Exponents
A positive exponent indicates how many times we multiply the base by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the positive exponent. This means we multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\). Understanding positive exponents is crucial as they are the foundation for more complex exponent rules.

When converting from negative to positive exponents, we use the reciprocal of the base, which we'll discuss in the next section.
Reciprocal
The reciprocal of a number is essentially one divided by that number. For instance, the reciprocal of 2 is \(\frac{1}{2}\). Negative exponents use this idea: they tell us to take the reciprocal of the base raised to the positive of that exponent.

In the given exercise, \((-3)^{-4}\) can be transformed by taking the reciprocal of \(-3\) and making the exponent positive: \((-3)^{-4} = \frac{1}{(-3)^4}\). By simplifying further, we multiply \(-3\) by itself four times.
Simplification
The process of simplification involves reducing an expression to its most basic form. After converting the negative exponent to a positive one and taking the reciprocal, we compute the positive exponent. For example, \((-3)^4\) means multiplying \(-3\) by itself four times: \(-3 \times -3 \times -3 \times -3 = 81\).

Then, \((-3)^{-4}\) simplifies to \(\frac{1}{81}\). The ultimate goal is to make expressions easier to work with and understand.

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Most popular questions from this chapter

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=x^{2}\\\ &g(x)=7 x-4 \end{aligned} $$

Interest Compounded Annually. An amount of money \(P\) that is invested at the yearly interest rate r grows to the amount $$ P(1+r)^{t} $$ after \(t\) years. Find a polynomial that can be used to determine the amount to which \(P\) will grow after 2 years.

Simplify. $$ \frac{\left(\frac{1}{2}\right)^{3}\left(\frac{2}{3}\right)^{4}}{\left(\frac{5}{6}\right)^{3}} $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 9^{7} \square 3^{13} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -16+5 $$
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