Problem 39 - Prealgebra

Question

Compute the exact square root. \(\sqrt{3.24}\)

Step-by-Step Solution

Answer

The square root of 3.24 is 1.8.

Step 1: Express as a Fraction

First, let's express the decimal number 3.24 as a fraction. The decimal 3.24 can be rewritten as \( \frac{324}{100} \). This is because 3.24 is equivalent to 324 hundredths.

Step 2: Simplify the Fraction

Next, simplify \( \frac{324}{100} \). Both the numerator and the denominator can be divided by 4, yielding \( \frac{81}{25} \).

Step 3: Apply the Square Root to Both Numerator and Denominator

Now, apply the square root to the simplified fraction. We have \( \sqrt{\frac{81}{25}} = \frac{\sqrt{81}}{\sqrt{25}} \).

Step 4: Calculate the Square Roots

Find the square roots of the numerator and denominator separately. The square root of 81 is 9, and the square root of 25 is 5. Thus, \( \frac{\sqrt{81}}{\sqrt{25}} = \frac{9}{5} \).

Step 5: Convert to Decimal Form

Finally, convert the fraction \( \frac{9}{5} \) back to decimal form. Dividing 9 by 5 gives us 1.8. Therefore, \( \sqrt{3.24} = 1.8 \).

Key Concepts

Fractions

Fractions are a powerful way to represent parts of a whole. They consist of a numerator and a denominator. The numerator is the top number, indicating how many parts we have, while the denominator is the bottom number, showing how many equal parts the whole is divided into. In this exercise, the number 3.24 is converted into a fraction. This step helps us manipulate the number more easily, especially when dealing with a square root.To express a decimal as a fraction:

• Identify the place value of the last digit. For 3.24, the 4 is in the hundredths place.
• Use that place value to form the fraction. Hence, 3.24 becomes \( \frac{324}{100} \).

Fractions are essential in mathematics, as they make calculations involving divisions, ratios, and proportions much easier to handle. Remembering how to convert decimals to fractions is a crucial foundation in math.

Simplifying Fractions

Simplifying fractions means breaking them down into their simplest form. This simplifies calculations and provides the most reduced way to express a numerical relationship.To simplify a fraction:

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

In our example, \( \frac{324}{100} \) is simplified by noticing both numbers are divisible by 4. Dividing by 4 gives us \( \frac{81}{25} \). Having a simplified fraction like \( \frac{81}{25} \) is important because it allows easier computation, especially when applying other mathematical operations such as finding square roots.

Decimal Conversion

Once we have a simplified fraction, we often convert it back into a decimal for easier interpretation, especially in practical situations.Decimal conversion involves:

• Dividing the numerator by the denominator.
• Using division to express the fraction as a decimal.

For instance, converting \( \frac{9}{5} \) to a decimal involves dividing 9 by 5, which gives 1.8. Decimal representation provides a straightforward understanding, particularly when dealing with percentages or other continuous quantities.