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Chapter 8: Problem 88

Simplify each expression. See Section 7.1 $$ \frac{9}{10}-\sqrt{\frac{49}{100}} $$

Short Answer

Expert verified
The expression simplifies to \( \frac{1}{5} \).

Step by step solution

01

Simplify the Square Root

First, simplify the square root in the expression. The expression inside the square root is \( \frac{49}{100} \). To simplify, take the square root of the numerator and the denominator separately:\[ \sqrt{\frac{49}{100}} = \frac{\sqrt{49}}{\sqrt{100}} \]Since \( \sqrt{49} = 7 \) and \( \sqrt{100} = 10 \), the expression simplifies to:\[ \frac{7}{10} \]
02

Subtract Fractions

Subtract the simplified square root \( \frac{7}{10} \) from \( \frac{9}{10} \). Since these fractions have the same denominator, subtract the numerators directly:\[ \frac{9}{10} - \frac{7}{10} = \frac{9 - 7}{10} \]Perform the subtraction:\[ \frac{2}{10} \]
03

Simplify the Fraction

Next, simplify the resulting fraction \( \frac{2}{10} \) by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 2. Divide both the numerator and the denominator by their GCD:\[ \frac{2 \div 2}{10 \div 2} = \frac{1}{5} \]Thus, the simplified expression is \( \frac{1}{5} \).

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Key Concepts

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

Square Roots
Square roots can simplify expressions significantly. When you encounter a square root in an expression, especially in a fractional form like \( \sqrt{\frac{49}{100}} \), it's helpful to simplify it step by step.
Start by separately taking the square roots of the numerator and the denominator. Here, \( \sqrt{49} \) equals 7 because 7 multiplied by itself gives 49. Similarly, \( \sqrt{100} \) equals 10. Therefore, \( \sqrt{\frac{49}{100}} \) simplifies to \( \frac{7}{10} \).
This approach not only makes the expression simpler but also sets us up for reliable arithmetic operations later, such as subtraction or addition.
Subtracting Fractions
When it comes to subtracting fractions, the process is straightforward if the denominators are the same. In our example, we had \( \frac{9}{10} - \frac{7}{10} \).
Since both fractions share the same denominator, you only need to subtract the numerators. So, subtract 7 from 9 to get 2, keeping the denominator as is. This gives us \( \frac{2}{10} \).
The logic is similar for addition: Maintain the common denominator and perform the operation on the numerators alone. This keeps the process simple and develops consistency in fraction operations.
Simplifying Fractions
Simplifying fractions is about expressing them in their simplest form. For the fraction \( \frac{2}{10} \), simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
The GCD of 2 and 10 is 2, so divide both by 2 to simplify. This results in \( \frac{1}{5} \). Simplifying helps in making calculations easier and gives a cleaner, more comprehensible result.
It's a good practice to always look for opportunities to simplify fractions in any math problem to avoid unnecessary complexity and errors.

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Most popular questions from this chapter

Neglecting air resistance, the distance \(s(t)\) in feet traveled by a freely falling object is given by the function \(s(t)=16 t^{2},\) where \(t\) is time in seconds. Use this formula to solve Exercises 79 through \(82 .\) Round answers to two decimal places. The Burj Khalifa, the tallest building in the world, was completed in 2010 in Dubai. It is estimated to be 2717 feet tall. How long would it take an object to fall to the ground from the top of the building? (Source: Council on Tall Buildings and Urban Habitat)

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find the \(y\) -intercept, approximate the \(x\) -intercepts to one decimal place, and sketch the graph. $$ f(x)=x^{2}-6 x+4 $$

Use the formula \(A=P(1+r)^{t}\) to solve Exercises 75 through 78 . See Example \(9 .\) Find the rate \(r\) at which \(\$ 3000\) compounded annually grows to \(\$ 4320\) in 2 years.

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=-4 x^{2}+8 x $$

If you are borrowing money at a rate of \(10 \%,\) would you prefer the interest to be simple or compound? Explain your answer.
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