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Chapter 7: Problem 26

Simplify. $$ \sqrt{0.36} $$

Short Answer

Expert verified
\( \sqrt{0.36} = \frac{3}{5} \)

Step by step solution

01

Identify the given expression

The given expression is \( \sqrt{0.36} \).
02

Convert decimal to fraction

\( 0.36 \) can be written as \( \frac{36}{100} \).
03

Express the fraction in simplest form

Simplify \( \frac{36}{100} \) by dividing both numerator and denominator by their greatest common divisor (4), which gives \( \frac{9}{25} \).
04

Apply the square root to the fraction

Take the square root of the simplified fraction: \( \sqrt{\frac{9}{25}} \).
05

Simplify the square root of the fraction

Simplify \( \sqrt{\frac{9}{25}} \) by taking the square root of the numerator and the denominator separately: \( \frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5} \).

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Key Concepts

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

Decimal to Fraction Conversion
When converting a decimal to a fraction, it's important to understand the place value of the decimal. For example, consider the decimal 0.36. The number 36 is in the hundredths place. Therefore, we can write it as \(\frac{36}{100}\) to represent the fraction. Since 36 is over 100, it simplifies the conversion process.
Remember:
  • Place the decimal number over its place value (e.g., hundredths, thousandths).
  • Eliminate the decimal by moving the given number multiple places to the right.
  • Simplify the fraction, if possible, by dividing both the numerator and the denominator by their common factors.
Following these steps will give you a straightforward way to convert any decimal to a fraction.
Greatest Common Divisor
To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can divide both numbers without leaving a remainder. For instance, in the fraction \(\frac{36}{100}\), both the numerator 36 and the denominator 100 have common factors:
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
The greatest common factor is 4. Therefore, we divide both the numerator and the denominator by 4: \(\frac{36 \div 4}{100 \div 4} = \frac{9}{25}\).
This process ensures that the fraction is in its simplest form.
Square Root of a Fraction
Finding the square root of a fraction involves taking the square root of both the numerator and the denominator separately. Consider the fraction \(\frac{9}{25}\). To find the square root:
  • Calculate the square root of the numerator: \(\sqrt{9} = 3\).
  • Calculate the square root of the denominator: \(\sqrt{25} = 5\).
Then, the square root of the fraction \(\frac{9}{25}\) is \(\frac{3}{5}\).
Always remember:
  • Simplify the fraction first, if necessary.
  • Apply the square root to both parts of the fraction separately.
  • If both the numerator and the denominator are perfect squares, the fraction will be straightforward to simplify.
This approach makes the square root calculation easily manageable.

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Most popular questions from this chapter

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