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Chapter 11: Problem 87

Simplify each expression. See Section 10.1 $$ \frac{3}{5}+\sqrt{\frac{16}{25}} $$

Short Answer

Expert verified
\( \frac{7}{5} \)

Step by step solution

01

Simplify the Square Root

First, simplify the square root in the expression \( \sqrt{\frac{16}{25}} \). The square root of a fraction is the square root of its numerator divided by the square root of its denominator. So, calculate \( \sqrt{16} \) and \( \sqrt{25} \).
02

Compute the Square Roots

Calculate \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \). Thus, \( \sqrt{\frac{16}{25}} = \frac{4}{5} \).
03

Add the Fractions

Now, add \( \frac{3}{5} \) and \( \frac{4}{5} \). Both fractions have the same denominator, so simply add the numerators: \( 3 + 4 = 7 \).
04

Write the Result

Combine the answer from the numerators over the common denominator: \( \frac{7}{5} \). This is the simplified expression.

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is a process of making a fraction as simple as possible. It involves finding the greatest common divisor (GCD) of both the numerator and the denominator and dividing each by their GCD.
  • A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.
  • For example, a fraction like \( \frac{3}{6} \) can be simplified by dividing both 3 and 6 by their GCD, which is 3, resulting in \( \frac{1}{2} \).
When fractions share a common denominator, as we see in this exercise, you can focus directly on adding or subtracting the following fractions.
Square Roots
A square root is a value that, when multiplied by itself, gives the original number. Finding the square root of a fraction might seem intimidating at first, but it's a straightforward process if you break it down:
  • The square root of a fraction \( \frac{a}{b} \) is the square root of the numerator \( a \) divided by the square root of the denominator \( b \).
  • For instance, with the fraction \( \frac{16}{25} \), you find \( \sqrt{16} \) and \( \sqrt{25} \) separately. Since \( 4 \times 4 = 16 \) and \( 5 \times 5 = 25 \), \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \).
By simplifying square roots in this way, it becomes much easier to handle them in more comprehensive expressions like the one given in the exercise.
Addition of Fractions
Adding fractions is easiest when the fractions involved have the same denominator. When you encounter these kinds of problems:
  • Add the numerators together while keeping the common denominator unchanged.
  • So, \( \frac{3}{5} + \frac{4}{5} = \frac{3+4}{5} = \frac{7}{5} \).
In the case where fractions have different denominators, you must first find a common denominator.
This often involves finding the least common multiple (LCM) of the denominators and converting each fraction to an equivalent fraction with the shared denominator before adding the numerators together.
Once added, check if the resulting fraction can be simplified, as seen previously with simplifying fractions.

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Most popular questions from this chapter

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ g(x)=-\frac{3}{2}(x-1)^{2}-5 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ g(x)=x^{2}+3 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ g(x)=x^{2}+7 $$

The quadratic function \(f(x)=668.7 x^{2}-2990.7 x+938\) approximates the U.S. growth of cell phone subscribers between 1985 and 2005 where \(x\) is the number of years past 1985 and \(f(x)\) is the number of subscribers in thousands. a. Use this function to approximate the number of subscribers in 2004 b. Use this function to predict the number of subscribers in 2007

Sketch the graph of each function. See Section 11.5 $$ g(x)=x+2 $$
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