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Chapter 3: Problem 90

Evaluate each expression. $$ \frac{5}{8} \cdot \frac{5}{8} $$

Short Answer

Expert verified
25/64

Step by step solution

01

Write down the expression

The given expression is \( \frac{5}{8} \times \frac{5}{8} \).
02

Multiply the numerators

Multiply the numerators of the fractions: \( 5 \times 5 = 25 \).
03

Multiply the denominators

Multiply the denominators of the fractions: \( 8 \times 8 = 64 \).
04

Combine the results

Combine the results to form the new fraction: \( \frac{25}{64} \).

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Key Concepts

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

Multiplying Fractions
When multiplying fractions, you essentially perform two straightforward multiplications鈥攐ne for the numerators and one for the denominators.
To multiply fractions \(\frac{a}{b} \times \frac{c}{d}\), simply follow these steps:
  • Multiply the numerators (top numbers) together: \(a \times c\).
  • Multiply the denominators (bottom numbers) together: \(b \times d\).
For example, given the expression \(\frac{5}{8} \times \frac{5}{8}\):
  • Multiply the numerators: \(5 \times 5 = 25\).
  • Multiply the denominators: \(8 \times 8 = 64\).
Therefore, the result is the fraction \(\frac{25}{64}\).
This process allows you to efficiently combine fractions through multiplication.
Numerators and Denominators
Understanding numerators and denominators is crucial for any fraction-related operations.
A fraction is composed of two parts:
  • The Numerator: The top number, indicating how many parts we have.
  • The Denominator: The bottom number, indicating the total number of equal parts the whole is divided into.
For instance, in the fraction \(\frac{5}{8}\):
  • The numerator is 5.
  • The denominator is 8.
When multiplying fractions, you need to treat these parts separately. Multiply the numerators together to create a new numerator, and multiply the denominators together to create a new denominator.
This separation helps simplify the multiplication process.
Simplification of Fractions
After multiplying fractions, you may often need to simplify the result to its lowest terms.
Simplifying a fraction involves dividing the numerator and the denominator by their greatest common divisor (GCD).
In our example \(\frac{25}{64}\), we need to check if the fraction can be simplified:
  • The numerator is 25.
  • The denominator is 64.
To simplify, find the GCD of 25 and 64, which is 1 since they have no common factors other than 1.
Since the GCD is 1, \(\frac{25}{64}\) is already in its simplest form.
Not all fractions require simplification, but when they do, it ensures the fraction is as straightforward as possible.

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Most popular questions from this chapter

The table shows the rate (in percent) at which 2-year college students (public) completed a degree within 3 years. $$ \begin{array}{|c|c|} \hline \text { Year } & {\text { Percent }} \\ \hline 2002 & {31.6} \\ {2003} & {30.1} \\ {2004} & {29.0} \\ {2005} & {27.5} \\ {2006} & {26.6} \\ {2007} & {26.9} \\ \hline \end{array} $$ (a) Write the data from the table as ordered pairs \((x, y),\) where \(x\) represents the year and \(y\) represents the percent. (b) What does the ordered pair \((2007,26.9)\) mean in the context of this problem? (c) Make a scatter diagram of the data, using the ordered pairs from part (a) and the given grid. (d) Describe the pattern indicated by the points on the scatter diagram. What is happening to rates at which 2 -year college students complete a degree within 3 years?

A point \((x, y)\) has the property that \(x y<0 .\) In which quadrant(s) must the point lie? Explain.

Write an equation for each line passing through the given point and having the given slope. Give the final answer in slope-intercept form. $$ (8,4), m=1 $$

Graph each equation by using the slope and y-intercept. $$ y=-\frac{1}{3} x+4 $$

Fill in each blank with the word positive or the word negative. The point with coordinates \((x, y)\) is in quadrant III if \(x\) is _____ and \(y\) is _____.
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