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Chapter 0: Problem 9

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(\frac{14}{21}\left(\frac{2}{5-\frac{1}{3}}\right)^{2}\)

Short Answer

Expert verified
The simplified form is \(\frac{6}{49}\).

Step by step solution

01

Simplify the Fractions

First, we simplify the fraction \(\frac{14}{21}\). Both 14 and 21 can be divided by 7, the greatest common divisor. Dividing the numerator and the denominator by 7 gives us \(\frac{14}{21} = \frac{2}{3}\).
02

Simplify the Denominator of Inner Fraction

Now, simplify the denominator of the inner fraction \(5 - \frac{1}{3}\). First, express 5 as \(\frac{15}{3}\) to have a common denominator with \(\frac{1}{3}\). So, \(5 - \frac{1}{3} = \frac{15}{3} - \frac{1}{3} = \frac{14}{3}\).
03

Simplify the Inner Fraction

Substitute the simplified denominator back into the inner fraction: \(\frac{2}{5-\frac{1}{3}} = \frac{2}{\frac{14}{3}}\). Dividing by a fraction is equivalent to multiplying by its reciprocal, so \(\frac{2}{\frac{14}{3}} = 2 \times \frac{3}{14} = \frac{6}{14}\). Simplify \(\frac{6}{14}\) by dividing both numerator and denominator by 2 to get \(\frac{3}{7}\).
04

Square the Simplified Fraction

Square the result from the previous step: \(\left(\frac{3}{7}\right)^2 = \frac{9}{49}\).
05

Multiply by the Simplified Outer Fraction

Finally, multiply the simplified outer fraction by the squared result: \(\frac{2}{3} \times \frac{9}{49} = \frac{2 \times 9}{3 \times 49} = \frac{18}{147}\).
06

Simplify the Final Fraction

Simplify \(\frac{18}{147}\) by finding the greatest common divisor, which is 3. Dividing the numerator and the denominator by 3 gives us \(\frac{6}{49}\).

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor (GCD) is a key concept in simplifying fractions. It represents the largest number that divides evenly into two or more numbers. To simplify a fraction like \( \frac{14}{21} \), we need to find the GCD of 14 and 21.
  • First, list the factors of both numbers.
  • For 14, the factors are 1, 2, 7, and 14.
  • For 21, the factors are 1, 3, 7, and 21.
The largest factor common to both is 7. Thus, dividing 14 and 21 by their GCD, 7, simplifies the fraction to \( \frac{2}{3} \). Using the GCD allows us to reduce fractions to their simplest form, making calculations easier and cleaner. Using the GCD is one of the most efficient ways to simplify fractions before further operations, such as multiplication or exponentiation.
Fraction Multiplication
Multiplying fractions might seem tricky at first, but it boils down to a simple set of steps.
  • Multiply the numerators together to get a new numerator.
  • Multiply the denominators together to get a new denominator.
When simplifying our expression, after simplifying fractions inside parenthesis, we get fractions that we need to multiply.In the example \( \frac{2}{3} \times \left( \frac{3}{7} \right)^2 \), after squaring \( \frac{3}{7} \) as \( \frac{9}{49} \), it requires multiplying \( \frac{2}{3} \) with \( \frac{9}{49} \). The multiplication becomes \( \frac{2 \times 9}{3 \times 49} = \frac{18}{147} \). After obtaining this result, it's crucial to simplify again. Understanding and practicing fraction multiplication helps in many mathematical areas and problem solving.
Exponents
Exponents are a way of expressing repeated multiplication. When you have a number or fraction with an exponent, you're multiplying it by itself that many times. For instance, \( \left( \frac{3}{7} \right)^2 \) means \( \frac{3}{7} \times \frac{3}{7} \). This results in:
  • Numerator: \( 3 \times 3 = 9 \)
  • Denominator: \( 7 \times 7 = 49 \)
This simplifies further calculations because it converts a power operation into a simple fraction multiplication. When applying exponents to fractions, ensure you square both the numerator and the denominator individually. This approach will help keep your expressions accurate when manipulating terms. In more advanced calculations, handling exponents with care is vital for maintaining accuracy.

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Most popular questions from this chapter

An open box is made by cutting squares of side \(x\) inches from the four corners of a sheet of cardboard 24 inches by 32 inches and then turning up the sides. Express the volume \(V(x)\) in terms of \(x\). What is the domain for this function?

Describe the natural domain of each function. (a) \(f(x)=\frac{x}{x^{2}-1}\) (b) \(g(x)=\sqrt{4-x^{2}}\)

Sketch the graph of each function. (a) \(f(x)=x^{2}-1\) (b) \(g(x)=\frac{x}{x^{2}+1}\) (c) \(h(x)=\left\\{\begin{array}{ll}x^{2} & \text { if } 0 \leq x \leq 2 \\\ 6-x & \text { if } x>2\end{array}\right.\)

Write \(F(x)=\sqrt{1+\sin ^{2} x}\) as the composite of four functions, \(f \circ g \circ h \circ k\).

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{4}+y^{4}=16\)
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