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Chapter 4: Problem 15

Write each expression using a positive exponent. $$5^{-3}$$

Short Answer

Expert verified
\(5^{-3}\) with a positive exponent is \(\frac{1}{5^3}\) or \(\frac{1}{125}\).

Step by step solution

01

Identify the Negative Exponent

The given expression is \(5^{-3}\). The exponent is negative, indicating that the base, 5, is in the denominator of a fraction.
02

Rewrite Using a Positive Exponent

To convert the negative exponent to a positive exponent, move the base from the numerator to the denominator of a fraction. This changes \(5^{-3}\) to \(\frac{1}{5^3}\), thus expressing the base with a positive exponent.
03

Simplify the Expression Further if Needed

Since the task is only to write the expression using a positive exponent, \(\frac{1}{5^3}\) satisfies the requirement. However, should further simplification be necessary, calculate \(5^3 = 5 \times 5 \times 5 = 125\), resulting in \(\frac{1}{125}\).

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Key Concepts

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

Understanding Exponents
Exponents are a way of representing repeated multiplication of a number by itself. When you see a number written as something like \(a^b\), \(a\) is called the base, and \(b\) is the exponent or power. The exponent tells you how many times to multiply the base by itself. For example, \(3^2\) means "3 multiplied by itself 2 times," which equals 9.
Exponents are a powerful tool for writing large numbers or simplifying multiplication. Instead of writing \(3 \times 3 \times 3\), you can simply write \(3^3\). There are different kinds of exponents, including positive and negative exponents, which we will explore in the following sections.
Simplifying Expressions with Exponents
When simplifying expressions with exponents, the aim is to make them easier to work with while keeping them mathematically equivalent. When dealing with a negative exponent, it can be helpful to convert it to a positive exponent.
  • With a negative exponent, such as \(a^{-b}\), you can rewrite it as \(\frac{1}{a^b}\).
  • This conversion helps to simplify expressions, especially in math problems involving fractions.

Another component of simplification is combining like terms. If you see exponents with the same base being multiplied together, like \(a^2 \times a^3\), you can add the exponents and simplify it to \(a^{2+3} = a^5\).
This rule applies when the bases are the same and you are multiplying. Simplification ensures expressions are in their simplest form for easier evaluation and understanding.
Transforming Negative Exponents into Positive Exponents
Negative exponents might look a bit intimidating at first, but they're straightforward to handle. A negative exponent indicates that the base is on the wrong side of a fraction line.
To transform it into a positive exponent, simply take the reciprocal of the base. For example, \(5^{-3}\) becomes \(\frac{1}{5^3}\).
Here is a strategy to manage negative exponents:
  • Identify if your exponent is negative.
  • To convert it, understand that a negative sign means "move the base to the other part of the fraction."
  • Write the base under 1 if it's on top, or write it on top if it was under, and change the sign of the exponent to positive.

Working through such conversions ensures that you can effectively handle and simplify expressions involving exponents without getting confused by the negative sign.

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Most popular questions from this chapter

OPEN ENDED Write a numerical fraction and an algebraic fraction in simplest form and a numerical fraction and an algebraic fraction not in simplest form.

Which One Doesn't Belong? Identify the fraction that does not belong with the other three. Explain your reasoning. $$\frac{6 y}{5}$$ $$\frac{4}{7}$$ $$\frac{1}{x^{2}}$$ $$\frac{15}{12}$$

Numbers can also be expressed in expanded form. Example \(1: 13,548=10,000+3000+500+40+8\) \(=\left(1 \times 10^{4}\right)+\left(3 \times 10^{3}\right)+\left(5 \times 10^{2}\right)+\left(4 \times 10^{1}\right)+\left(8 \times 10^{0}\right)\) Example \(2: 0.568=0.5+0.06+0.008\) $$ =\left(5 \times 10^{-1}\right)+\left(6 \times 10^{-2}\right)+\left(8 \times 10^{-3}\right) $$ Write each number in expanded form. $$0.173$$

The weight of a fruit fly is about \(1.3 \times 10^{-4}\) pound. How many pounds would one million fruit flies weigh?

Simplify each fraction. If the fraction is already in simplest form, write simplified. $$\frac{12 m}{15 m}$$
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