Problem 33 - Prealgebra

Question

In Exercises 33-52, compute the exact square root. \(\sqrt{0.81}\)

Step-by-Step Solution

Answer

The square root of 0.81 is 0.9.

Step 1: Identify the Expression

We need to find the square root of 0.81, which is written mathematically as \( \sqrt{0.81} \).

Step 2: Express as a Fraction

Realize that 0.81 can be expressed as a fraction: \( 0.81 = \frac{81}{100} \). This conversion can help in computing the square root.

Step 3: Separate the Square Roots

Use the property of square roots that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). So, \( \sqrt{0.81} = \sqrt{\frac{81}{100}} = \frac{\sqrt{81}}{\sqrt{100}} \).

Step 4: Compute the Square Roots Separately

Compute the square root of the numerator and the denominator. \( \sqrt{81} = 9 \) and \( \sqrt{100} = 10 \).

Step 5: Simplify the Result

Plug the square roots into the fraction: \( \frac{\sqrt{81}}{\sqrt{100}} = \frac{9}{10} \).

Step 6: Convert Fraction to Decimal

Convert \( \frac{9}{10} \) back to a decimal, which is 0.9.

Key Concepts

Fraction Conversion

Converting a decimal to a fraction is a helpful step in solving mathematical problems, such as finding square roots. It involves expressing a number with a decimal point as a fraction where the numerator and the denominator are integers. This makes further computations, like finding square roots, more manageable.

To convert a decimal to a fraction, follow these steps:

• Identify the place value of the last digit. For 0.81, the last digit "1" is in the hundredths place.
• Write the decimal number as a fraction with 1 under it: \[ 0.81 = \frac{81}{100} \]
• By expressing 0.81 as \( \frac{81}{100} \), it becomes easier to handle, especially when working with square roots.

Converting decimals to fractions lays the groundwork for further operations such as finding square roots or simplifying the expression.

Decimals

Decimals are a way of expressing numbers that are not whole, using a decimal point to separate the whole part from the fractional part. They are widely used in everyday contexts and scientific calculations.

Each digit after the decimal point represents a "power of ten" fraction:

• The first digit is tenths (0.1), the second is hundredths (0.01), the third is thousandths (0.001), and so forth.
• For example, 0.81 means 8 tenths and 1 hundredth, making its total equivalent to \( \frac{81}{100} \).

Decimals are convenient for representing parts of whole numbers, making them a crucial aspect of arithmetic and algebraic calculations. They are also easily converted to fractions, allowing flexible approaches in problem-solving, such as finding square roots like \( \sqrt{0.81} \). When you convert between decimals and fractions, it opens up different strategies to reach your solution. Decimals inherently give a more visual and intuitive impression of the size of a number.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This makes calculations easier and solutions more understandable. When a fraction is in its simplest form, the numerator and denominator have no common factors other than 1.

To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

Taking the fraction \( \frac{81}{100} \) as an example:

• The GCD of 81 and 100 is 1, meaning \( \frac{81}{100} \) is already simplified.
• However, when taking square roots, \( \frac{\sqrt{81}}{\sqrt{100}} \) gives us \( \frac{9}{10} \), a straightforward simplified form.

Simplifying the result of square root operations or any other fractions guarantees clarity and often ease of interpretation. Keeping expressions simple not only aids in calculations but also ensures clear communication of results. Simplified expressions are frequently used in final solutions to express results clearly and concisely.