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Chapter 4: Problem 66

Find each product or quotient. Express using exponents. $$\frac{5^{5}}{5^{2}}$$

Short Answer

Expert verified
The product is \( 5^3 \).

Step by step solution

01

Identify the Expression

We need to find the quotient of two exponents: \( \frac{5^{5}}{5^{2}} \). This expression involves division of powers with the same base.
02

Apply the Division Rule for Exponents

The division rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). In this case, \( a = 5 \), \( m = 5 \), and \( n = 2 \). Applying the rule gives:\[\frac{5^5}{5^2} = 5^{5-2}\]
03

Simplify the Exponent

Subtract the exponents as indicated:\[5^{5-2} = 5^3\]
04

Final Answer

The simplified expression using exponents is \( 5^3 \).

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Key Concepts

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

Division Rule for Exponents
When dealing with exponents, the division rule is incredibly handy. It is used to simplify expressions where exponents with the same base are divided. This rule states that when you divide two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator.
For example, if you have \( \frac{a^m}{a^n} \), it simplifies to \( a^{m-n} \). This rule is essential because it allows us to transform what seems like a complex fraction into a much simpler expression with a single power. This makes calculations easier, particularly when dealing with large numbers.
  • Example: \( \frac{5^5}{5^2} = 5^{5-2} = 5^3 \)
  • Key Tip: Always ensure the bases are the same before applying the division rule.
Simplifying Exponents
After applying the division rule for exponents, the resulting expression often requires simplification. Simplifying exponents means reducing the expression to its most compact form. In the case of the example \( 5^{5-2} \), once the subtraction is done, you obtain \( 5^3 \).
Simplification is the process of actually calculating or interpreting \( 5^3 \) as the operation of 5 multiplied by itself three times, which equals 125. However, when the question or task asks to express the answer in exponential form, you can leave it as \( 5^3 \)
  • Why Simplify: Simplification makes mathematical expressions easier to work with, especially when performing additional operations.
  • In Practice: While expressing terms in their simplest exponential form is often required, understanding what it means in terms of repeated multiplication is beneficial.
Quotient of Powers
The term "quotient of powers" might sound technical, but it refers to dividing one exponent by another. As explained earlier, the division rule for exponents simplifies such divisions by subtracting the exponents, given that the base is the same.
This concept is crucial for handling expressions with multiple exponents efficiently. Understanding the quotient of powers allows you to manage, simplify, and compute large numbers that would otherwise be cumbersome.
  • Essential Idea: Reducing fractions involving exponents by subtracting their powers can simplify equations quickly.
  • Remember: Only works when the bases of the powers involved in the quotient are identical.

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Most popular questions from this chapter

Which store offers the best buy? \begin{aligned}&\begin{array}{l}\text { A A } \\\\\text { B B } \\\\\text { C } C \\\\\text { D D }\end{array}\\\&\begin{array}{|c|c|}\hline \text { Store } & \text { Potatoes } \\\\\hline \mathrm{A} & 18 \text { for } \$ 12 \\\\\hline \mathrm{B} & 30 \text { for } \$ 24 \\\\\hline \mathrm{C} & 36 \text { for } \$ 30 \\\\\hline \mathrm{D} & 42 \text { for } \$ 30 \\\\\hline\end{array}\end{aligned}

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Find each product. $$1.6 \times 1000$$

Find each product. $$0.05 \times 1000$$

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