Problem 39 Simplify. $$\left(\frac{1}{3}-... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 4: Problem 39

Simplify. $$\left(\frac{1}{3}-\frac{1}{2}\right)^{2}$$

Short Answer

Expert verified

$\frac{1}{36}$

Step by step solution

Subtract the fractions inside the parentheses

To simplify $\frac{1}{3} - \frac{1}{2}$, find a common denominator for the fractions. The common denominator for 3 and 2 is 6. Rewrite each fraction with the common denominator: $\frac{1}{3} = \frac{2}{6}$ and $\frac{1}{2} = \frac{3}{6}$. So, the expression becomes $\frac{2}{6} - \frac{3}{6}$.

Perform the subtraction

Subtract the fractions: $\frac{2}{6} - \frac{3}{6} = -\frac{1}{6}$.

Square the result

Now, take $-\frac{1}{6}$ and square it: $\bigg(-\frac{1}{6}\bigg)^2$. Squaring a negative fraction involves squaring both the numerator and the denominator, yielding $\frac{(-1)^2}{6^2} = \frac{1}{36}$.

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Key Concepts

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

Common Denominator

When dealing with fractions, one vital step is finding the common denominator. A common denominator helps you add or subtract fractions by making the denominators the same. For example, in the fraction operation $\frac{1}{3} - \frac{1}{2}$, you need to find a number that both 3 and 2 can divide without any remainders. In this case, the smallest common multiple is 6.
To convert each fraction to have this common denominator:

Rewrite $\frac{1}{3}$ as $\frac{2}{6}$.

Rewrite $\frac{1}{2}$ as $\frac{3}{6}$.

By aligning the denominators, you can now easily perform addition or subtraction.

Fraction Subtraction

With a common denominator, subtracting fractions becomes straightforward. Let's continue with our example using $\frac{2}{6}$ and $\frac{3}{6}$.
To subtract fractions:

Subtract the numerators: \2 - 3 = -1\.
Keep the denominator the same: 6. Therefore, $\frac{2}{6} - \frac{3}{6} = -\frac{1}{6}$.

This result shows that subtracting fractions is similar to subtracting whole numbers, but with an added step of ensuring the denominators match.

Squaring Fractions

The final step in our example is squaring the result of the fraction subtraction. We have \- \frac{1}{6}\ from our previous operation, so now we need to square it.
Squaring a fraction means raising both the numerator and the denominator to the power of 2: \left( -\frac{1}{6} \right)^2\.

$ -1 $^2 = 1\ because the square of any negative number is positive.
And, \6^2 = 36\.

So, \left( -\frac{1}{6} \right)^2 = \frac{1}{36}\.
Remember this when you square a fraction:

Square both the numerator and the denominator independently.

This completes simplifying \left( \frac{1}{3} - \frac{1}{2} \right)^2\.

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91影视

Simplify. $$\left(\frac{1}{3}-\frac{1}{2}\right)^{2}$$

Short Answer

Step by step solution

Subtract the fractions inside the parentheses

Perform the subtraction

Square the result

Key Concepts

Common Denominator

Fraction Subtraction

Squaring Fractions

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