Problem 17 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplification of \( \frac{18}{\sqrt{162}} \) is \( \sqrt{2} \).

Step 1: Rationalize the Denominator

To simplify \( \frac{18}{\sqrt{162}} \), we need to eliminate the square root in the denominator by multiplying both the numerator and the denominator by \( \sqrt{162} \):\[\frac{18}{\sqrt{162}} \times \frac{\sqrt{162}}{\sqrt{162}} = \frac{18\sqrt{162}}{162}\]

Step 2: Simplify the Denominator

Next, simplify the denominator. Since \( 162 = 81 \times 2 \) and \( \sqrt{81} = 9 \), we have:\[162 = (9)^2 \times 2, \quad \text{thus, } \sqrt{162} = 9\sqrt{2}\]Replace \( \sqrt{162} \) in the expression:\[\frac{18\sqrt{162}}{162} = \frac{18 \times 9\sqrt{2}}{162}\]

Step 3: Simplify the Fraction

Now simplify the fraction. Notice that \( 18 \times 9 = 162 \), so:\[\frac{18 \times 9\sqrt{2}}{162} = \frac{162\sqrt{2}}{162} = \sqrt{2}\]

Step 4: Conclusion

The simplified form of the expression \( \frac{18}{\sqrt{162}} \) is \( \sqrt{2} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a technique used in algebra to eliminate any square roots or irrational numbers from the bottom of a fraction. This is done to make the expression simpler and easier to work with. When we have a fraction where the denominator contains a square root, like in the example \( \frac{18}{\sqrt{162}} \), the aim is to "rationalize" it. To achieve this, we multiply both the numerator and the denominator by the square root present in the denominator. This helps in getting an integer or a more workable expression in the denominator.

For instance:

• Multiply the fraction \( \frac{18}{\sqrt{162}} \) by \( \frac{\sqrt{162}}{\sqrt{162}} \) to get: \( \frac{18\sqrt{162}}{162} \).
• This step ensures the square root is eliminated from the denominator.
• Rationalizing the denominator is a fundamental step to simplify expressions and solve algebra problems effectively.

Simplifying Expressions

Simplifying expressions in algebra involves reducing complex fractions or numerical expressions to their simplest form. This process makes it easier to understand and use the expressions in further calculations or equations. The key to simplifying expressions like \( \frac{18\sqrt{162}}{162} \) is to look for common factors in the numerator and denominator.

Some steps to keep in mind include:

• Factoring numbers to their prime components can reveal common factors.
• Use these factors to reduce fractions to their simplest terms.
• In our example, since \( 18 \times 9 = 162 \), the fraction simplifies by canceling these common factors.
• After simplification, the fraction \( \frac{162\sqrt{2}}{162} \) condenses down to \( \sqrt{2} \).

By following these simple strategies, you can streamline expressions into more convenient forms.

Square Roots in Algebra

Square roots appear often in algebra, and understanding how they function is key to solving many problems. A square root essentially asks "what number, when multiplied by itself, results in a given number?" For example, \( \sqrt{81} = 9 \) because \( 9 \times 9 = 81 \).

In expressions like \( \sqrt{162} \), finding a simpler form is a valuable skill:

• First, factor the number under the square root to find perfect squares. In this case, \( 162 = 81 \times 2 \).
• The square root of 81 is 9, simplifying the root to \( 9\sqrt{2} \).
• This breaking down of roots can help in solving and simplifying equations.
• Understanding the properties of square roots allows for easier manipulation of algebraic expressions.

Overall, mastering square roots in algebra empowers you to tackle equations and expressions with confidence.