Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Rationalize the denominators for the given expressions. Assume all expressions containing \(x\) are positive. $$\frac{3}{2 \sqrt{10}}$$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{3 \sqrt{10}}{20} \).

Step 1: Identify the Denominator Issue

The expression to simplify is \( \frac{3}{2 \sqrt{10}} \). The denominator contains a square root, which needs to be rationalized to make it a rational number.

Step 2: Multiply by the Conjugate

To eliminate the square root in the denominator, multiply both the numerator and the denominator by \( \sqrt{10} \), which is the radical in the denominator. This gives: \( \frac{3 \cdot \sqrt{10}}{2 \sqrt{10} \cdot \sqrt{10}} \).

Step 3: Simplify Sqrt Multiplication

Calculate \( \sqrt{10} \cdot \sqrt{10} \), which is \( 10 \), so the denominator becomes \( 2 \times 10 = 20 \). The expression is now \( \frac{3 \sqrt{10}}{20} \).

Step 4: Final Simplification

The expression \( \frac{3 \sqrt{10}}{20} \) cannot be simplified further since 3, 10, and 20 have no common factors besides 1.

Key Concepts

Simplifying Radicals

When we talk about radicals, we're usually dealing with square roots. Simplifying radicals means making the expression as simple as possible. For instance, if you have \( \sqrt{50} \), you can break it down into \( \sqrt{25 \times 2} \). Since \(25\) is a perfect square, it becomes \( 5 \). So, \( \sqrt{50} \) simplifies to \( 5\sqrt{2} \).

Here are some tips to simplify radicals:

• Look for perfect square factors within the radical.
• Factorize the number under the radical to its prime factors.
• Simplify repeatedly until no more perfect squares can be factored out.

Once you've simplified a radical, it becomes easier to work with, especially in complicated expressions.

Conjugate in Algebra

Conjugates are a useful tool in algebra to rationalize denominators and eliminate radicals. The conjugate of a binomial expression involves changing the sign between two terms. For example, the conjugate of \( a + b \sqrt{c} \) would be \( a - b \sqrt{c} \).

To use conjugates for rationalizing, multiply both the numerator and denominator of a fraction by the conjugate of the denominator. This process helps turn any radical expressions in the denominator into a rational number. When multiplying conjugates \((a + b \sqrt{c})(a - b \sqrt{c})\), the middle terms cancel out due to the difference of squares, leaving \(a^2 - b^2c\).

This technique not only helps in simplifying expressions but also makes it possible to further simplify calculations involving complex numbers, polynomials, or irrational denominators.

Square Roots in Denominator

Rationalizing the denominator is about changing an expression so the denominator becomes a rational number, without a square root. Consider the expression \( \frac{3}{2 \sqrt{10}} \). Here, the denominator is \( 2 \sqrt{10} \), which is an irrational number.

To rationalize, multiply the entire fraction by \( \sqrt{10} \) both in the numerator and denominator. The idea is to use the property \( \sqrt{a} \cdot \sqrt{a} = a \) to eliminate the square root. So, \( 2 \sqrt{10} \cdot \sqrt{10} = 20 \), which simplifies the expression to a rational number in the denominator. After this operation, the fraction becomes \( \frac{3 \sqrt{10}}{20} \).

This method ensures that the denominator is rationalized, making the expression simpler to interpret and calculate with. It's an essential skill in algebra to handle expressions involving radicals more effectively.