Problem 27 Use the order of operations to s... [FREE SOLUTION]

Chapter 4: Problem 27

Use the order of operations to simplify each expression. $$\left(\frac{3}{4}+\frac{1}{8}\right)^{2}-\left(\frac{1}{2}+\frac{1}{8}\right)$$

Short Answer

Expert verified

The expression simplifies to $ \frac{9}{64} $.

Step by step solution

Simplify Inside the Parentheses

First, solve the expression inside the parentheses, starting with $ \left(\frac{3}{4} + \frac{1}{8}\right) $. Find a common denominator: $ \frac{3}{4} = \frac{6}{8} $. So, $ \frac{6}{8} + \frac{1}{8} = \frac{7}{8} $.

Simplify the Second Parentheses

Now, simplify the second parentheses: $ \left(\frac{1}{2} + \frac{1}{8}\right) $. A common denominator is also $ 8 $: $ \frac{1}{2} = \frac{4}{8} $. Thus, $ \frac{4}{8} + \frac{1}{8} = \frac{5}{8} $.

Apply the Exponent

Use the result from Step 1 and apply the exponent: $ \left(\frac{7}{8}\right)^2 = \frac{7}{8} \times \frac{7}{8} = \frac{49}{64} $.

Subtract the Expressions

From Step 3's result, subtract the result from Step 2: $ \frac{49}{64} - \frac{5}{8} $. Convert $ \frac{5}{8} $ to $ \frac{40}{64} $ for a common denominator. Calculate: $ \frac{49}{64} - \frac{40}{64} = \frac{9}{64} $.

Conclusion

The simplified expression is $ \frac{9}{64} $. Follow these steps to ensure accuracy and adherence to the order of operations.

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Key Concepts

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

Fractions

Fractions represent a part of a whole. They are written in the form of $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator. Understanding fractions is crucial in mathematics as they regularly appear in various operations.
Here are some quick points about fractions:

The numerator tells us how many parts we have.
The denominator tells us how many parts make up a whole.
Fractions can be simplified by dividing both the numerator and denominator by their greatest common divisor.

In the original exercise, fractions are added and subtracted. Remember to find a common denominator first before addition or subtraction. This aligns the parts by making sure they refer to the same-sized segments.

Exponents

Exponents are a way to express repeated multiplication. When you see something like $a^b$, it means that $a$ is multiplied by itself $b$ times.
Here鈥檚 what to keep in mind about exponents:

$a^1 = a$, because any number to the power of one is itself.
$a^0 = 1$, because any number to the power of zero equals one.
An exponent outside of parentheses means you must apply it to everything inside.

In our original step-by-step solution, the exponent applies to the fraction inside the parentheses, turning $\left(\frac{7}{8}\right)^2$ into $\frac{49}{64}$. This essential step must follow the order of operations.

Common Denominators

When dealing with fractions, having a common denominator simplifies addition or subtraction. This is because fractions with the same denominator are easier to compare and combine. To find a common denominator, you can:

Identify the least common multiple (LCM) of the denominators.
Convert each fraction so their denominators are equal to this LCM.

In the exercise, the fractions $\frac{3}{4}$ and $\frac{1}{8}$ are converted to have a common denominator of 8: $\frac{6}{8}$ and $\frac{1}{8}$, which add up seamlessly. Finding common denominators is a fundamental step in handling fractions computationally.

Parentheses in Math

Parentheses indicate which operations should be performed first in a mathematical expression. This is crucial, especially when mixed with other operations like multiplication, division, addition, or subtraction.
Key points to remember:

Solve inside the parentheses first before any operations outside.
Work from the innermost parentheses outward in complex expressions.
Parentheses can change the outcome of calculations, altering the order of operations.

In the exercise, parentheses dictate that the expression $\left(\frac{3}{4} + \frac{1}{8}\right)$ should be simplified before applying any exponent. This underscores the importance of following the order of operations for accurate results.

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91影视

Use the order of operations to simplify each expression. $$\left(\frac{3}{4}+\frac{1}{8}\right)^{2}-\left(\frac{1}{2}+\frac{1}{8}\right)$$

Short Answer

Step by step solution

Simplify Inside the Parentheses

Simplify the Second Parentheses

Apply the Exponent

Subtract the Expressions

Conclusion

Key Concepts

Fractions

Exponents

Common Denominators

Parentheses in Math

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