Problem 195 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{1}{4} \sqrt{98}-\frac{1}{3} \sqrt{128} $$

Step-by-Step Solution

Answer

\( \frac{-11\sqrt{2}}{12} \)

Step 1: Simplify the square roots

First, simplify the square roots of 98 and 128. We have: \( \ \sqrt{98} = \sqrt{2 \times 49} = \sqrt{2 \times 7^2} = 7\sqrt{2} \ \ \sqrt{128} = \sqrt{2^7} = \sqrt{2^6 \times 2} = 8 \sqrt{2} \)

Step 2: Substitute the simplified forms

Substitute \(\sqrt{98}\) and \(\sqrt{128}\) with their simplified forms: \( \ \frac{1}{4} 7\sqrt{2} - \frac{1}{3} 8\sqrt{2} \ = \frac{7\sqrt{2}}{4} - \frac{8\sqrt{2}}{3} \)

Step 3: Find a common denominator

Find a common denominator for the fractions. The common denominator of 4 and 3 is 12: \( \ \frac{7\sqrt{2}}{4} = \frac{7\sqrt{2} \times 3}{4 \times 3} = \frac{21\sqrt{2}}{12} \ \ \frac{8\sqrt{2}}{3} = \frac{8\sqrt{2} \times 4}{3 \times 4} = \frac{32\sqrt{2}}{12} \)

Step 4: Subtract the fractions

Now, subtract the fractions: \( \ \frac{21\sqrt{2}}{12} - \frac{32\sqrt{2}}{12} = \frac{21\sqrt{2} - 32\sqrt{2}}{12} = \frac{-11\sqrt{2}}{12} \)

Key Concepts

Square Roots

Square roots are used to find a number which, when multiplied by itself, equals the original number. For example, \( \sqrt{16} \ = 4\), since \( 4 \times 4 \ = 16\). Simplifying square roots involves breaking them down into their prime factors and extracting squares. In the exercise, we simplified \( \sqrt{98} \) into \( 7 \sqrt{2}\) and \( \sqrt{128} \) into \( 8 \sqrt{2}\). This step is crucial because it allows us to work with simpler expressions.

Fraction Subtraction

Subtracting fractions involves a few steps to ensure the calculations are correct. We need to make the denominators the same, so we can then subtract the numerators. In the exercise, we had two fractions: \( \frac{7\sqrt{2}}{4}\) and \( \frac{8\sqrt{2}}{3}\). The fractions were first brought to a common denominator before subtracting their numerators.

Common Denominator

Finding a common denominator is necessary when we want to add or subtract fractions. The common denominator for two fractions is the least common multiple (LCM) of their denominators. In the exercise, we found the common denominator for 4 and 3 by using the LCM, which is 12. This allowed us to rewrite both fractions with a common base: \( \frac{7\sqrt{2}}{4} = \frac{21\sqrt{2}}{12}\) and \( \frac{8\sqrt{2}}{3} = \frac{32\sqrt{2}}{12}\).

Simplifying Expressions

In mathematics, simplifying expressions makes them easier to work with. This often involves combining like terms, factoring, and other methods to rewrite the expression in a simpler form. In the given exercise, after finding a common denominator, we subtracted the numerators directly: \( \frac{21\sqrt{2}}{12} - \frac{32\sqrt{2}}{12} = \frac{-11\sqrt{2}}{12}\). Thus, our final simplified expression became \( \frac{-11\sqrt{2}}{12}\).