Problem 42 Simplify the following. $$\lef... [FREE SOLUTION]

Chapter 4: Problem 42

Simplify the following. $$\left(-\frac{3}{4}\right)^{2}+\frac{3}{8}$$

Short Answer

Expert verified

$ \frac{15}{16} $

Step by step solution

Square the Fraction

Start by squaring the fraction inside the parentheses: $ \left(-\frac{3}{4}\right)^2 $. To do this, square both the numerator and the denominator: $ (-3)^2 = 9 $ and $ (4)^2 = 16 $. Therefore, $ \left(-\frac{3}{4}\right)^2 = \frac{9}{16} $.

Add the Second Fraction

Now add $ \frac{9}{16} $ and $ \frac{3}{8} $. To do this, you must first have a common denominator for the fractions. The least common denominator of 16 and 8 is 16. Convert $ \frac{3}{8} $ to have a denominator of 16: $ \frac{3}{8} = \frac{3 \times 2}{8 \times 2} = \frac{6}{16} $.

Perform the Addition

Now add $ \frac{9}{16} $ and $ \frac{6}{16} $, since they have a common denominator: $ \frac{9}{16} + \frac{6}{16} = \frac{9 + 6}{16} = \frac{15}{16} $. The simplified result is $ \frac{15}{16} $.

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Key Concepts

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

Squaring Fractions

When squaring a fraction, your goal is to multiply it by itself. Let鈥檚 break down the meaning:

Simplify the expression $-\frac{3}{4}$: here, both the numerator and the denominator are involved in the squaring process.
Multiply $-3$ by $-3$, which gives you $9$. Negative times negative results in a positive.
Multiply $4$ by $4$, yielding $16$.

This turns $\left(-\frac{3}{4}\right)^2$ into $\frac{9}{16}$. Squaring negatives might seem confusing, but just remember, the negative sign disappears as it becomes positive after squaring.

Common Denominator

Before you can add fractions like $\frac{9}{16}$ and $\frac{3}{8}$, they need to have the same denominator. This is important because you can only add fractions when they 'speak the same language' so to speak.

Identify the least common denominator (LCD): for 16 and 8, this is 16.
Convert $\frac{3}{8}$ to have this common denominator: multiply both the numerator and the denominator by $2$ to get $\frac{6}{16}$.

Now that both fractions have the denominator 16, you're set to add them together.

Fraction Addition

Adding fractions becomes easy when we've found a common denominator. Here鈥檚 how it works:

Once you have $\frac{9}{16}$ and $\frac{6}{16}$, simply add the numerators.
This results in $9 + 6 = 15$.
The denominator remains the same: $16$.

Thus, the resulting fraction is $\frac{15}{16}$. It's important to check if your final fraction can be simplified. In this case, $\frac{15}{16}$ is already in its simplest form as 15 and 16 have no common factors besides 1. So, you are done!

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91影视

Simplify the following. $$\left(-\frac{3}{4}\right)^{2}+\frac{3}{8}$$

Short Answer

Step by step solution

Square the Fraction

Add the Second Fraction

Perform the Addition

Key Concepts

Squaring Fractions

Common Denominator

Fraction Addition

One App. One Place for Learning.

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