Problem 17 - College Algebra

Question

Simplify each expression. $$\frac{18}{\sqrt{162}}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the square root

We need to simplify the square root in the denominator, \( \sqrt{162} \). Start by finding the prime factorization of 162, which is \( 162 = 2 \times 3^4 \). Thus, \( \sqrt{162} = \sqrt{2 \times 3^4} = \sqrt{2} \times 3^2 \). We can simplify this further as \( 3^2 \sqrt{2} = 9\sqrt{2} \).

Step 2: Substitute the simplified square root into the expression

Now, substitute \( 9\sqrt{2} \) for \( \sqrt{162} \) in the expression. Thus, the expression becomes \( \frac{18}{9\sqrt{2}} \).

Step 3: Simplify the fraction

Simplify the fraction \( \frac{18}{9} \) by dividing both the numerator and the denominator by 9. This gives us \( \frac{2}{\sqrt{2}} \).

Step 4: Rationalize the denominator

To rationalize the denominator of \( \frac{2}{\sqrt{2}} \), multiply both the numerator and the denominator by \( \sqrt{2} \). This gives us \( \frac{2\sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2} \).

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. To begin simplifying an expression, particularly when dealing with roots, prime factorization can be incredibly useful.

To find the prime factors, break the number down into smaller parts until you reach the prime numbers. Let's take 162 as an example. The first prime factor of 162 is 2, so we divide 162 by 2 to get 81.

Next, we continue breaking down 81. Since 81 is not divisible by 2, we move to the next smallest prime number, which is 3. Dividing 81 by 3 repeatedly, we find that 81 equals \(3 \times 3 \times 3 \times 3\), or \(3^4\).

So, the complete prime factorization of 162 is \(2 \times 3^4\). Prime factorization simplifies calculations, especially with square roots, making further computations easier.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is a critical step in simplifying expressions, ensuring they are in a standard mathematical form.

In our example expression, \(\frac{2}{\sqrt{2}}\), the denominator has a square root, \(\sqrt{2}\). To rationalize it, multiply both the numerator and the denominator by the same square root. This keeps the value of the fraction the same.

• Multiply \(\frac{2}{\sqrt{2}}\) by \(\frac{\sqrt{2}}{\sqrt{2}}\).
• This results in \(\frac{2\sqrt{2}}{\sqrt{2} \times \sqrt{2}}\).

Remember that \(\sqrt{2} \times \sqrt{2}\) results in 2, because multiplying two identical square roots gives you the number under the root. Thus, you get \(\frac{2\sqrt{2}}{2}\), which simplifies further to \(\sqrt{2}\). This result is a cleaner, rational representation of the expression.

Square Roots

Understanding square roots is fundamental in many areas of mathematics. A square root of a number is a value which, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because \(2 \times 2 = 4\).

When simplifying expressions involving square roots, one effective approach is to look for perfect squares within the number. This step helps in breaking down the expression more effectively.

For instance, when dealing with \(\sqrt{162}\), using its prime factorization helps us see that it includes \(3^4\), or \((3^2)^2\). Since \(3^2\) is a perfect square, it can be taken out of the square root as 9, leaving us with \(9\sqrt{2}\).

• The process involves: identifying perfect squares in the factorization.
• Simplifying by taking these out of the root.

Being comfortable with square roots and how they simplify can make handling complex expressions much more manageable.