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Chapter 4: Problem 391

In the following exercises, simplify. $$\frac{\left(\frac{1}{3}\right)^{2}}{5+2^{2}}$$

Short Answer

Expert verified
\(\frac{1}{81}\)

Step by step solution

01

Simplify the numerator

First, simplify the numerator \ \ \left(\frac{1}{3}\right)^{2} \ \ Square the fraction inside the parentheses: \(\left(\frac{1}{3}\right)^{2} = \frac{1^2}{3^2} = \frac{1}{9}\)
02

Simplify the denominator

Now simplify the denominator \ \ 5 + 2^2 \ \ First, calculate \(2^2\): \(2^2 = 4\) \ \ Add this result to 5: 5 + 4 = 9
03

Divide the simplified numerator by the simplified denominator

Now divide \(\frac{1}{9}\) by 9 \ \ This can also be written as \(\frac{1}{9} \div 9\) \ \ Which is equivalent to \(\frac{1}{9} \times \frac{1}{9} = \frac{1 \times 1}{9 \times 9} = \frac{1}{81}\)

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Key Concepts

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

Exponents
Exponents are a way to show how many times a number, called the base, is multiplied by itself. For example, in \(2^3\), 2 is the base, and 3 is the exponent. This means 2 is multiplied by itself three times: \(2 \times 2 \times 2 = 8\). In our exercise, we have \(2^2\). This simplifies to 4 because \(2 \times 2 = 4\).
Recall a key rule: \( (a/b)^n = \frac{a^n}{b^n}\). This helps us simplify expressions with exponents and fractions effectively.
Fractions
Fractions consist of a numerator (top number) and a denominator (bottom number). Simplifying fractions means reducing them to their simplest form. For instance, to simplify \( \frac{1}{3} \), we need to square the fraction as given in the exercise: \((\frac{1}{3})^{2} \). This means you square the numerator and the denominator separately: \ \frac{1^2}{3^2} = \frac{1}{9} \.
This methodology ensures each part of the fraction is handled properly, simplifying the entire process.
Order of Operations
The order of operations is essential in simplifying mathematical expressions properly. You can remember the correct order using PEMDAS:
  • P: Parentheses first
  • E: Exponents (ie Powers and Square Roots, etc.)
  • MD: Multiplication and Division (left-to-right)
  • AS: Addition and Subtraction (left-to-right)

In our exercise, we first handle the exponent in the denominator: \(2^2 = 4\). Next, we add that result to 5: 5 + 4 = 9. Finally, we handle the division of the simplified fraction by this result, following the process carefully to ensure accuracy.

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Most popular questions from this chapter

In the following exercises, solve the equation. $$ \frac{m}{-2}=-16 $$

In the following exercises, change to equivalent fractions using the given LCD. $$\frac{1}{3} \text { and } \frac{1}{5}, \quad L C D=15$$

In the following exercises, perform the indicated operation. $$ 6 \frac{2}{5}+7 \frac{3}{5} $$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{a+b}{a-b}\) when \(a=-3, b=8\)

In the following exercises, solve the equation. $$ h-\left(-\frac{7}{8}\right)=-\frac{2}{5} $$
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