Problem 12 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{\frac{27}{64}} $$

Step-by-Step Solution

Answer

\( \frac{3\sqrt{3}}{8} \)

Step 1: Simplify the square root

Start by simplifying the expression under the square root. Notice that the expression is a fraction, and the square root can be taken for both the numerator and the denominator individually: \[ \sqrt{\frac{27}{64}} = \frac{\sqrt{27}}{\sqrt{64}}. \]

Step 2: Simplify the denominator

Calculate the square root of the denominator, 64. Since 64 is a perfect square, we find \[ \sqrt{64} = 8. \]

Step 3: Simplify the numerator

Now simplify the square root of the numerator, 27, by factoring it into a perfect square: \[ 27 = 9 \times 3 = 3^2 \times 3. \] Thus, \[ \sqrt{27} = \sqrt{3^2 \times 3} = 3\sqrt{3}. \]

Step 4: Combine simplified results

Combine the simplified results from steps 2 and 3 to get the simplified expression: \[ \frac{\sqrt{27}}{\sqrt{64}} = \frac{3\sqrt{3}}{8}. \]

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They allow us to find a number which, when multiplied by itself, gives the original number. When dealing with square roots of fractions, as in the expression \( \sqrt{\frac{27}{64}} \), we can simplify by considering the square root of the numerator and the denominator separately. Here are a few key points to remember:

• For any positive number \( x \), the square root is denoted as \( \sqrt{x} \).
• The process involves looking for perfect squares (numbers like 1, 4, 9, 16, etc.) within the number in the square root.
• If dealing with fractions, simplify using \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

This approach helps in breaking down and simplifying expression, making calculations easier.

Factoring Expressions

Factoring expressions can make simplifying radicals much more straightforward. In the case of the expression \( \sqrt{27} \), we need to break it down into factors to identify if any part is a perfect square. Here's how you can do it:

• Identify components of the number that can be rewritten as squares. For instance, \( 27 = 9 \times 3 \). Here, 9 is a perfect square, \( 9 = 3^2 \).
• Expressing it in terms of powers, \( 27 = 3^2 \times 3 \), clarifies which part can exit the square root.
• Remember that \( \sqrt{3^2 \times 3} = 3\sqrt{3} \), because the square root of \( 3^2 \) is 3.

By understanding factoring, you simplify inside the root first, allowing you to easily pull perfect squares out from under the root sign.

Rational Expressions

Rational expressions simplify the process of working with fractions, especially when they involve radicals. To manage expressions like \( \frac{\sqrt{27}}{\sqrt{64}} \), consider these steps:

• First, ensure the fraction is in its simplest form by simplifying the square roots involved separately.
• Identify the perfect square in the denominator; since \( 64 \) is \( 8^2 \), \( \sqrt{64} = 8 \).
• With the numerator simplified to \( 3\sqrt{3} \), the expression is \( \frac{3\sqrt{3}}{8} \).
• This clean simplification means your radical is now wrapped up in a neat rational expression for clarity.

Simplifying radicals into rational expressions makes calculations manageable and results more understandable.