Problem 341 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{96}}{\sqrt{150}} $$

Step-by-Step Solution

Answer

\[\frac{4}{5}\]

Step 1: Simplify the Numerator

Simplify the expression under the square root in the numerator. Since \(96 = 16 \times 6\), and \(16\) is a perfect square, it can be simplified as follows: \[\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}\]

Step 2: Simplify the Denominator

Simplify the expression under the square root in the denominator. Since \(150 = 25 \times 6\), and \(25\) is a perfect square, it can be simplified as follows: \[\sqrt{150} = \sqrt{25 \times 6} = \sqrt{25} \times \sqrt{6} = 5\sqrt{6}\]

Step 3: Simplify the Fraction

Now, substitute the simplified forms of the numerator and denominator into the original fraction: \[\frac{\sqrt{96}}{\sqrt{150}} = \frac{4\sqrt{6}}{5\sqrt{6}}\] Since \(\sqrt{6}\) appears in both the numerator and the denominator, they cancel each other out: \[\frac{4\sqrt{6}}{5\sqrt{6}} = \frac{4}{5}\]

Key Concepts

square roots

A square root is a number that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 because 4 multiplied by 4 equals 16. Square roots are often represented with the radical symbol: \(\sqrt{}\). When simplifying square roots, look for perfect squares, which are numbers like 1, 4, 9, 16, 25, etc.

These numbers are squares of integers and help make square roots simpler. For example, \(\sqrt{16 \times 6}=4\sqrt{6} \). Instead of dealing with a large number like 96, we simplify it using its factors.

fractions

A fraction represents a part of a whole. The top number (numerator) and the bottom number (denominator) show how the whole is divided. For example, in \( \frac{4}{5} \), 4 is the numerator and 5 is the denominator.

When simplifying fractions, look for common factors to cancel out. In our example, \( \frac{\sqrt{96}}{\sqrt{150}}=\frac{4\sqrt{6}}{5\sqrt{6}} \). Since \(\sqrt{6}\) is present in both the numerator and the denominator, it cancels out, leaving \( \frac{4}{5} \).

perfect squares

Perfect squares are numbers obtained by squaring whole numbers. Examples include 1 (1x1), 4 (2x2), 9 (3x3), 16 (4x4), and so on. Perfect squares play a crucial role in simplifying square roots.

For instance, in simplifying \( \sqrt{96} \), we recognize that 16 (which is 4x4) is a factor, so \( \sqrt{96} \) becomes \( \sqrt{16 \times 6}=4\sqrt{6} \). Similarly, for \( \sqrt{150} \), we factorize it to \( \sqrt{25 \times 6 }=5\sqrt{6} \). By simplifying using perfect squares, we make the calculation easier.