Problem 23 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \sqrt{\frac{49}{225}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\sqrt{\frac{49}{225}}$. Answer: $\frac{7}{15}$.

Step 1: Distribute the square root

First, let's distribute the square root across the fraction. $$ \sqrt{\frac{49}{225}} = \frac{\sqrt{49}}{\sqrt{225}} $$

Step 2: Simplify square roots

Now, we will simplify the square roots for each of the perfect squares. $$ \frac{\sqrt{49}}{\sqrt{225}} = \frac{7}{15} $$ So, the simplified expression is: $$ \sqrt{\frac{49}{225}} = \frac{7}{15} $$

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics that help simplify expressions involving squares. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, since \(7 \times 7 = 49\), the square root of 49 is 7.

Represented by the radical symbol \(\sqrt{}\), square roots can be distributed across multiplication and division inside the radical. This property allows us to simplify complex expressions, like a square root of a fraction.

Understanding square roots will enhance your ability to break down and simplify mathematical problems. This concept is vital for algebra and calculus, where simplifying expressions is often required.

Breaking Down Fractions

Fractions represent a part of a whole and consist of a numerator (top part) and a denominator (bottom part). They can appear in various math problems, such as equations and expressions. When dealing with fractions in square roots, it's essential to remember that you can separate the square root of a fraction into two separate roots — one for the numerator and one for the denominator.

• The operation looks like this: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
• Use this property to simplify your expression by simplifying each part separately.

Breaking down fractions into simpler parts not only makes calculations more manageable but also enhances your understanding of the underlying problem. As you practice separating square roots in fractions, you'll become more adept at simplifying complex mathematical expressions.

Perfect Squares Made Easy

A perfect square is a number that can be expressed as the product of an integer with itself. In other words, if a number is a perfect square, it can be squared to yield an exact integer. Recognizing perfect squares can speed up the process of simplifying square roots.

• Examples of perfect squares include 1, 4, 9, 16, 25, 36, 49, and so forth, as they are all squares of whole numbers.
• To simplify square roots, determine if the number inside the root is a perfect square. If it is, you can effortlessly find its square root.

For instance, \(\sqrt{49} = 7\), since 49 is a perfect square. This understanding helps when simplifying expressions where perfect squares appear under the root sign.

Familiarity with perfect squares will greatly simplify your calculations and lead to quicker problem-solving in various mathematical tasks.