Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\frac{18}{\sqrt{162}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \sqrt{2} \).

Step 1: Simplify the Denominator

First, simplify the square root in the denominator. The number 162 can be factored as follows: \[ 162 = 2 \times 81 = 2 \times 9^2. \] Thus, the simplified form of \( \sqrt{162} \) is \( \sqrt{2 \times 9^2} = 9\sqrt{2} \).

Step 2: Write the Expression with the Simplified Denominator

Now replace \( \sqrt{162} \) with \( 9\sqrt{2} \) in the original fraction. The expression becomes \[ \frac{18}{9\sqrt{2}}. \]

Step 3: Simplify the Fraction

Divide both the numerator and the denominator of the fraction \( \frac{18}{9\sqrt{2}} \) by 9 to simplify:\[ \frac{18 \div 9}{9\sqrt{2} \div 9} = \frac{2}{\sqrt{2}}. \]

Step 4: Rationalize the Denominator

To eliminate the square root from the denominator, multiply the numerator and the denominator by \( \sqrt{2} \). Doing so gives:\[ \frac{2 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{2}}{2}. \]

Step 5: Simplify the Final Expression

Since \( \frac{2\sqrt{2}}{2} = \sqrt{2} \), the simplified form of the expression is \( \sqrt{2} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is key to simplifying expressions with square roots in the denominator. The aim is to remove any irrational numbers from the denominator to make it a rational number. It helps in simplifying the expression further and is preferred in most mathematical representations.
To rationalize a denominator like \( \frac{2}{\sqrt{2}} \), we multiply both the numerator and the denominator by \( \sqrt{2} \). This technique works because multiplying two square roots like \( \sqrt{2} \times \sqrt{2} \) results in an integer, which is 2 in this case. The expression becomes:

• Numerator: \( 2 \times \sqrt{2} = 2\sqrt{2} \)
• Denominator: \( \sqrt{2} \times \sqrt{2} = 2 \)

After this step, the expression simplifies to \( \frac{2\sqrt{2}}{2} \), which allows us to cancel the 2s, leaving \( \sqrt{2} \).
This makes the denominator rational and neat, showcasing why rationalizing is a valuable tool in algebra.

Square Roots

Square roots are used to find a number which, when multiplied by itself, results in the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). In mathematics, square roots often appear in expressions that need simplification because they can help reduce more complex radicals to manageable forms.
When simplifying \( \sqrt{162} \), consider the prime factors of 162, which are:

• 162 = 2 × 81
• 81 is the square of 9

Thus, \( \sqrt{162} \) can be computed as \( \sqrt{2 \times 9^2} \), simplifying to \( 9\sqrt{2} \).
Breaking down square roots into simpler components makes many algebraic operations much easier and eliminates the complexities involved with square roots in expressions.

Mathematical Fractions

Fractions are a way to represent parts of a whole and are described by a numerator and a denominator. They require careful manipulation, especially when square roots are involved, as seen when simplifying expressions like \( \frac{18}{\sqrt{162}} \).
Initially, simplify the fraction by dividing both parts by their greatest common divisor. In \( \frac{18}{9\sqrt{2}} \), this is done by dividing by 9:

• Numerator: \( 18 \div 9 = 2 \)
• Denominator: \( 9\sqrt{2} \div 9 = \sqrt{2} \)

Thus, you get \( \frac{2}{\sqrt{2}} \).
By simplifying fractions first, further algebraic steps such as rationalizing the denominator become clearer, and expressions can be neatly presented. Mastery of fractions is foundational for progressing in math, as it simplifies problem-solving and enhances understanding of algebraic operations.