Calculating Square Root of Decimal Numbers

The problem asks us to evaluate the expression \(\sqrt {\frac{{0.064{\rm{\;}} \times {\rm{\;}}6.25}}{{0.081{\rm{\;}} \times {\rm{\;}}4.84}}} \). This involves calculating the square root of a fraction where both the numerator and the denominator are products of decimal numbers.

Converting Decimals to Fractions for Calculation

A helpful approach to simplify this expression is to convert each decimal number into its equivalent fractional form. This eliminates the decimals, making the calculation under the square root easier.

• The decimal 0.064 can be written as \(\frac{64}{1000}\).
• The decimal 6.25 can be written as \(\frac{625}{100}\).
• The decimal 0.081 can be written as \(\frac{81}{1000}\).
• The decimal 4.84 can be written as \(\frac{484}{100}\).

Substituting Fractions into the Expression

Now, let's substitute these fractions back into the original square root expression:

\( \sqrt {\frac{{\frac{{64}}{{1000}} \times \frac{{625}}{{100}}}}{{\frac{{81}}{{1000}} \times \frac{{484}}{{100}}}}} \)

We can rewrite the fraction inside the square root. The numerator is \(\frac{64 \times 625}{1000 \times 100}\) and the denominator is \(\frac{81 \times 484}{1000 \times 100}\). When dividing fractions, we multiply by the reciprocal of the denominator:

\( \sqrt {\frac{{64 \times 625}}{{1000 \times 100}} \times \frac{{1000 \times 100}}{{81 \times 484}}} \)

Notice that the term \(1000 \times 100\) appears in both the numerator and the denominator, so they cancel out:

\( \sqrt {\frac{{64 \times 625}}{{81 \times 484}}} \)

Finding the Square Root of Each Factor

Using the property that the square root of a product is the product of the square roots (\(\sqrt{ab} = \sqrt{a}\sqrt{b}\)) and the square root of a quotient is the quotient of the square roots (\(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)), we can write:

\( \sqrt {\frac{{64 \times 625}}{{81 \times 484}}} = \frac{{\sqrt{64} \times \sqrt{625}}}{{\sqrt{81} \times \sqrt{484}}} \)

Now, we calculate the square root of each individual number:

• \(\sqrt{64} = 8\)
• \(\sqrt{625} = 25\) (since \(25 \times 25 = 625\))
• \(\sqrt{81} = 9\)
• \(\sqrt{484} = 22\) (since \(22 \times 22 = 484\))

Completing the Multiplication and Simplification

Substitute the calculated square roots back into the expression:

\( \frac{{8 \times 25}}{{9 \times 22}} \)

Perform the multiplication in the numerator and the denominator:

Numerator: \(8 \times 25 = 200\)

Denominator: \(9 \times 22 = 198\)

So, the expression simplifies to the fraction \(\frac{200}{198}\).

Finally, we simplify the fraction \(\frac{200}{198}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

\( \frac{200 \div 2}{198 \div 2} = \frac{100}{99} \)

Result of the Calculation

The value of the given expression \(\sqrt {\frac{{0.064{\rm{\;}} \times {\rm{\;}}6.25}}{{0.081{\rm{\;}} \times {\rm{\;}}4.84}}} \) is \(\frac{100}{99}\).

Revision Table: Important Square Roots

Reviewing the square roots used in the calculation is useful for quick reference:

Additional Information: Strategies for Decimal Square Roots

Dealing with square roots of decimal numbers can be tricky, but converting them to fractions is a reliable method, especially when the numbers involved are perfect squares or can be easily reduced. This technique leverages the properties of fractions and square roots to simplify complex expressions.

Alternatively, you could notice the number of decimal places. In this problem, the total number of decimal places in the numerator (3 from 0.064 + 2 from 6.25 = 5) is equal to the total number of decimal places in the denominator (3 from 0.081 + 2 from 4.84 = 5). When this happens, you can effectively ignore the decimal points and place the numbers as integers inside the square root, because the powers of 10 needed to clear the decimals will cancel out:

\( \sqrt {\frac{{0.064 \times 6.25}}{{0.081 \times 4.84}}} = \sqrt {\frac{{64 \times 625}}{{81 \times 484}}} \times \sqrt{\frac{10^{-3} \times 10^{-2}}{10^{-3} \times 10^{-2}}} = \sqrt {\frac{{64 \times 625}}{{81 \times 484}}} \times \sqrt{\frac{10^{-5}}{10^{-5}}} = \sqrt {\frac{{64 \times 625}}{{81 \times 484}}} \times \sqrt{1} \)

This shortcut works because the decimal place counts match, leading to the same simplified expression under the root.