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Chapter 6: Problem 55

Simplify by starting at "the bottom" and working upward. $$ 7-\frac{3}{5+\frac{2}{4-2}} $$

Short Answer

Expert verified
\( \frac{13}{2} \)

Step by step solution

01

Simplify the innermost denominator

Look at the expression inside the innermost fraction: \( 4 - 2 \). Simplify this first.\[ 4 - 2 = 2 \]
02

Simplify the inner fraction

The innermost denominator has simplified to 2. Now simplify the fraction: \( \frac{2}{2} \).\[ \frac{2}{2} = 1 \]
03

Substitute the simplified inner fraction back

Replace the inner fraction in the larger expression with the simplified result:\[ 7 - \frac{3}{5 + 1} \]
04

Simplify the new denominator

Simplify the denominator \(5 + 1\):\[ 5 + 1 = 6 \]
05

Simplify the entire fraction

Substitute back the simplified denominator and simplify the fraction:\[ 7 - \frac{3}{6} \]Simplify \( \frac{3}{6} \):\[ \frac{3}{6} = \frac{1}{2} \]
06

Subtract the simplified fraction from 7

Subtract \( \frac{1}{2} \) from 7:\[ 7 - \frac{1}{2} = \frac{14}{2} - \frac{1}{2} = \frac{13}{2} \]

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Key Concepts

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

Nested Fractions
A nested fraction is a fraction that contains another fraction in its numerator or denominator. These can look tricky at first, but the key is to simplify them step by step, starting from the innermost fraction.

For instance, in the expression \(7 - \frac{3}{5+\frac{2}{4-2}}\), we have to handle the fraction within the fraction first.
Begin by simplifying the innermost fraction to manage the complexity. In this case, the innermost fraction is \( \frac{2}{4-2}\), where you simplify the expression \(4 - 2 \) to get \( \frac{2}{2} \).

Here's how you go about it:
  • Simplify the expression \(4 - 2 = 2 \)
  • Then, simplify the fraction \( \frac{2}{2} = 1 \)
  • Replace the inner fraction back into the larger expression already simplified: \( 5 + 1 \)
  • Continue simplifying as you move outward.

By breaking it down systematically, you can simplify even the most complex-looking nested fractions.
Order of Operations
Order of operations is crucial when working with nested fractions. It tells you which part of an expression to compute first. We use the acronym PEMDAS, which stands for: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Following PEMDAS ensures you simplify correctly. In the expression \(7 - \frac{3}{5+\frac{2}{4-2}}\), you start inside the parentheses:

  • Simplify \(4 - 2 = 2\)
  • Then, the fraction \( \frac{2}{2} = 1 \)
  • Next, add \(5 + 1 = 6 \)
  • Finally, replace into the larger expression: \(7 - \frac{3}{6} \)

By following the order of operations, you ensure every step results in a correct and simplified form.
Arithmetic Operations
Arithmetic operations are the basic

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Most popular questions from this chapter

Write four equivalent forms for each rational expression. $$ -\frac{2 x-9}{7 x-1} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{3 x^{2}-5 x-2}{x-2} \cdot \frac{x-3}{x+1} $$

Multiply. Write each answer in lowest terms. $$ \frac{4(y-2)}{x} \cdot \frac{3 x}{6(y-2)^{2}} $$

Multiply. Write each answer in lowest terms. $$ \frac{(a+b)^{2}}{5} \cdot \frac{30}{2(a+b)} $$

Divide. Write each answer in lowest terms. $$ \frac{4 t^{4}}{2 t^{5}} \div \frac{(2 t)^{3}}{-6} $$
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