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Chapter 9: Problem 2

Write each expression with positive exponents, then simplify. $$3^{-4}$$

Short Answer

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

Step by step solution

01

Understand Negative Exponents

A negative exponent, like in the expression \(3^{-4}\), indicates that the base is on the wrong side of a fraction line. To make the exponent positive, we need to take the reciprocal of the base and change the sign of the exponent.
02

Re-write with Positive Exponents

Convert the expression with a negative exponent to one with a positive exponent by taking the reciprocal of the base. The expression \(3^{-4}\) can be rewritten as \(\frac{1}{3^4}\).
03

Simplify the Expression

Now calculate \(3^4\). Since \(3^4 = 3 \times 3 \times 3 \times 3 = 81\), the expression becomes \(\frac{1}{81}\).

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Key Concepts

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

Understanding Reciprocals
A reciprocal is a key concept that helps us in dealing with negative exponents. Essentially, the reciprocal of a number is what you multiply it by to get a result of 1. For a simple number, like 2, the reciprocal is easy to find: just flip it over. So, the reciprocal of 2 is \(\frac{1}{2}\).
For more complex numbers or variables, reciprocals work the same way. Here's what to remember:
  • The reciprocal of \(x\) is \(\frac{1}{x}\).
  • Multiplying a number by its reciprocal gives you 1: \(x \times \frac{1}{x} = 1\).
  • To remove a negative exponent, use the reciprocal of the base.
In our example with \(3^{-4}\), the base is 3. To turn the negative exponent into a positive one, we take the reciprocal of 3, which is \(\frac{1}{3}\). This takes us halfway to our solution!
Simplifying Expressions
Simplifying expressions is about making them as straightforward as possible. When expressions involve exponents, the goal is to express them with positive exponents whenever possible. Here's a quick guide:
  • Identify parts of the expression with negative exponents.
  • Convert those parts by using their reciprocals, turning the negative exponents into positive ones.
  • Combine like terms and simplify calculations where possible.
In our example, after recognizing \(3^{-4}\) as a candidate for simplification, we convert it to \(\frac{1}{3^4}\). Now, it's ready for the next step - actually calculating the value of \(3^4\). This is a critical skill for clearer and more useful expressions.
The Power of Positive Exponents
Positive exponents appear often in math and are simpler to manage because you perform straightforward multiplications. Understanding how to handle them sets the stage for tackling more complex problems. Here's why positive exponents are awesome:
  • They indicate repeated multiplication of a base number.
  • They simplify expressions to easily understandable forms.
  • They are the standard way to express powers, making them integral in evaluations and reducing complexity.
In our example calculation from \(3^{-4}\), we transform it to \(\frac{1}{3^4}\), taking advantage of having a positive exponent. We multiply 3 by itself four times to get 81. This means the expression becomes \(\frac{1}{81}\). Working with positive exponents ensures consistency and clarity in mathematical operations.

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Most popular questions from this chapter

Multiply the following expressions. \(\left(6 x^{2} y^{3}\right)\left(-5 x^{3} y^{5}\right)\)

Multiply the following expressions. \(\left(7 a^{3} b^{2}\right)\left(8 a b^{3}\right)\)

Find the value of each of the following expressions when \(x=4\). $$-5 x+6$$

Find each quotient. Write all answers in scientific notation. $$\frac{0.00000047}{235,000}$$

Simplify each expression, and write all answers in scientific notation. $$\frac{\left(2 \times 10^{-3}\right)\left(6 \times 10^{-5}\right)}{3 \times 10^{-4}}$$
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