Problem 62 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{6 \sqrt{5}}{5 \sqrt{12}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{15}}{5}\)

Step 1: Simplify the Denominator

The denominator is \(5 \sqrt{12}\). Simplify \(\sqrt{12}\) by expressing 12 as a product of a perfect square: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}\). Therefore, the denominator becomes \(5 \times 2 \sqrt{3} = 10 \sqrt{3}\).

Step 2: Rationalize the Denominator

To eliminate the radical in the denominator of \(\frac{6 \sqrt{5}}{10 \sqrt{3}}\), multiply the numerator and the denominator by \(\sqrt{3}\). This gives \(\frac{6 \sqrt{5} \times \sqrt{3}}{10 \sqrt{3} \times \sqrt{3}} = \frac{6 \sqrt{15}}{10 \times 3} = \frac{6 \sqrt{15}}{30}\).

Step 3: Simplify the Fraction

Now simplify the fraction \(\frac{6 \sqrt{15}}{30}\). Divide both the numerator and the denominator by 6: \(\frac{6 \sqrt{15}}{30} = \frac{\sqrt{15}}{5}\).

Step 4: Result in Simplest Radical Form

The expression is now in its simplest radical form: \(\frac{\sqrt{15}}{5}\). There are no more perfect squares in the radical, and the fraction is fully simplified.

Key Concepts

Rationalizing Denominators

Rationalizing denominators is a technique used to eliminate radicals, such as square roots, from the bottom (denominator) of a fraction. Having radicals in the denominator is generally not favored in mathematics. Hence, rationalizing is an important step to know.

To rationalize the denominator, we multiply both the numerator and the denominator by a radical that will help cancel out the radical present in the denominator. For example, in the fraction \( \frac{6 \sqrt{5}}{10 \sqrt{3}} \), we have a \( \sqrt{3} \) in the denominator. We can eliminate this by multiplying the entire fraction by \( \sqrt{3}/\sqrt{3} \).

• This results in \( \frac{6 \sqrt{5} \times \sqrt{3}}{10 \sqrt{3} \times \sqrt{3}} \).
• The denominator becomes \( 10 \times 3 \), which is 30, a rational number.

The process simplifies to a fraction with no radicals in the denominator: \( \frac{6 \sqrt{15}}{30} \). This is how rationalizing simplifies the expression and makes it more convenient to work with.

Simplifying Radicals

Simplifying radicals is the process of finding an equivalent expression for a radical expression that has no perfect square factors other than 1 inside the square root. This makes the expression easier to work with or interpret.

Let's consider \( \sqrt{12} \). The number 12 can be broken down as a product of a perfect square, specifically, \( 4 \times 3 \).

• The square root of 4 is a perfect square, \( 2 \), simplifying \( \sqrt{4} \times \sqrt{3} \) into \( 2\sqrt{3} \).
• Now \( \sqrt{12} \) in simplest radical form is \( 2 \sqrt{3} \).

Notice how we've transformed the original radical expression into one with simpler components. This technique reduces the complexity and thus helps in subsequent steps when simplifying fractions or expressions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. These can include variables, numbers, and radicals, making them an essential topic in algebra.

When working with algebraic fractions, it is often necessary to simplify the fraction. Here's how it's done in the given example:

• Start with the expression \( \frac{6 \sqrt{15}}{30} \).
• Both the numerator and the denominator must be simplified: We notice that 6 and 30 have a common factor, which is 6.
• Dividing both the numerator and the denominator by 6 gives \( \frac{\sqrt{15}}{5} \).

Simplifying algebraic fractions involves cancelling common factors whenever possible, streamlining the expressions for easier computation and interpretation. This approach not only finds a simpler form of the expression but also helps maintain balance and equality within algebraic computations.