Problem 12 - Algebra A Combined Function

Question

Find each square root. Assume that all variables represent nonnegative real numbers. $$ \sqrt{0.04} $$

Step-by-Step Solution

Answer

\( \sqrt{0.04} = 0.2 \)

Step 1: Rewrite the Decimal as a Fraction

The square root can be more easily calculated if the decimal number is expressed as a fraction. Rewrite 0.04 as a fraction with a power of 10: \( 0.04 = \frac{4}{100} \).

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{4}{100} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives \( \frac{1}{25} \).

Step 3: Find the Square Root of the Simplified Fraction

Use the property of square roots for fractions: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). So, \( \sqrt{\frac{1}{25}} = \frac{\sqrt{1}}{\sqrt{25}} = \frac{1}{5} \).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them can make further calculations easier. This involves reducing the fraction to its simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD). Let's break it down.
Suppose you have the fraction \( \frac{4}{100} \). Here, 4 is the numerator, and 100 is the denominator.

• First, find the GCD of 4 and 100. The GCD is the largest number that divides both 4 and 100 without leaving a remainder. In this case, the GCD is 4.
• Next, divide both the numerator and the denominator by the GCD. So, \( \frac{4}{100} \) becomes \( \frac{4 \div 4}{100 \div 4} = \frac{1}{25} \).

By simplifying the fraction, we make it easier to work with, especially when performing operations like finding square roots. So, always look to simplify a fraction before proceeding with other calculations.

Decimals to Fractions

Converting decimals to fractions involves a straightforward process and can make calculations easier. To illustrate, let's consider the decimal 0.04.
First, understand that every decimal can be expressed as a fraction where the denominator is a power of ten. Here's how you can convert a decimal like 0.04 to a fraction:

• Count the number of decimal places. In 0.04, there are two decimal places.
• Write 0.04 as \( \frac{4}{100} \). This step involves removing the decimal point and placing the number over a denominator that is 1 followed by as many zeros as there are decimal places. Hence, it's 100 because of the two decimal places.

Writing decimals as fractions helps in operating them mathematically, such as simplifying or finding roots or other manipulative tasks, with ease. This step is crucial for tasks like simplifying or solving equations involving square roots, where fractions are often more manageable than decimals.

Properties of Square Roots

Square roots have properties that help simplify expressions and solve equations. For instance, when you have a fraction, the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) is particularly useful. Here's how it works in practice:
Given the fraction \( \frac{1}{25} \), and you need to find its square root:

• Apply the square root property: \( \sqrt{\frac{1}{25}} = \frac{\sqrt{1}}{\sqrt{25}} \).
• Calculate the square root of the numerator and the denominator individually. For \( \sqrt{1} \), the answer is 1, because 1 is a perfect square. For \( \sqrt{25} \), the answer is 5, as 5 times itself equals 25.
• Thus, \( \sqrt{\frac{1}{25}} = \frac{1}{5} \).

The rules for square roots allow for breaking down complex expressions into smaller, more manageable parts. This property not only simplifies calculations but also provides a clear strategy to approach such problems efficiently.