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Chapter 10: Problem 27

Find each square root, if possible. $$\sqrt{\frac{81}{25}}$$

Short Answer

Expert verified
The square root of \(\frac{81}{25}\) simplifies to \(\frac{9}{5}\).

Step by step solution

01

Identify the Numerator and Denominator

We can see that the given fraction is \(\frac{81}{25}\), so the numerator is 81 and the denominator is 25.
02

Check if Both Numbers are Perfect Squares

A perfect square is a number that can be expressed as the product of another number with itself. In this case, both 81 and 25 are perfect squares: - \(81 = 9 \times 9\) - \(25 = 5 \times 5\) We can find the square root of each number separately.
03

Find the Square Roots

Now that we know both the numerator and the denominator are perfect squares, we can find their square roots: - \(\sqrt{81} = 9\) - \(\sqrt{25} = 5\)
04

Simplify the Fraction

Now, we can simplify the fraction by substituting the square roots back into the original expression: \(\sqrt{\frac{81}{25}} = \frac{\sqrt{81}}{\sqrt{25}} = \frac{9}{5}\) Therefore, the square root of \(\frac{81}{25}\) simplifies to \(\frac{9}{5}\).

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Key Concepts

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

Perfect Squares
A perfect square is a number that results from multiplying an integer by itself. These numbers are special because they have whole numbers as their square roots. Understanding perfect squares can make calculating square roots much easier. For example, the numbers 81 and 25 from the original exercise are perfect squares. This is because:
  • 81 can be expressed as 9 multiplied by 9, or \(9 \times 9\).
  • 25 is 5 multiplied by 5, or \(5 \times 5\).
Knowing that 81 and 25 are perfect squares means they each have integer square roots, namely 9 and 5, respectively. Thus, recognizing perfect squares helps simplify calculations by avoiding complex decimal roots.
Fraction Simplification
Simplifying fractions involves reducing the fraction to its simplest form, which often makes solving mathematical expressions easier. When dealing with square roots, simplifying a fraction can involve finding the square roots of both the numerator and the denominator separately. In the example,
  • The fraction \(\frac{81}{25}\) is given.
  • The square roots are \(\sqrt{81} = 9\) and \(\sqrt{25} = 5\).
  • The simplified fraction is obtained as \(\frac{9}{5}\).
By simplifying the fraction in this manner, we move from a complex expression to a simpler one that is easier to understand and use in further calculations.
Numerator and Denominator
In any fraction, the numerator and the denominator are crucial components. The numerator is the top part and represents how many parts we are considering. The denominator is the bottom part and shows into how many parts the whole is divided.
  • For an example like \(\frac{81}{25}\), the numerator is 81 and the denominator is 25.
  • Both numbers must be carefully considered, especially when finding square roots of fractions, to ensure calculations are correct.
  • Finding the square roots separately (\(\sqrt{81}\) and \(\sqrt{25}\)) helps in simplifying the expression.
Understanding the roles of both the numerator and denominator helps in performing accurate and efficient mathematical operations with fractions.

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Most popular questions from this chapter

Simplify completely. Assume all variables represent positive real numbers. $$\sqrt[5]{x^{5} y^{15}}$$

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Multiply and simplify. $$\sqrt{\frac{6}{5}} \cdot \sqrt{\frac{1}{8}}$$

Rationalize the denominator of each expression. $$\sqrt{\frac{18}{26}}$$

Fill in the blank. Assume all variables represent positive real numbers. $$\sqrt[3]{p} \cdot \sqrt[3]{7}=\sqrt[3]{p^{3}}=p$$
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