Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression by rationalizing the denominator. Simplify your answer.\(\frac{5}{2 \sqrt{10}-5}\)

Step-by-Step Solution

Answer

The expression \(\frac{5}{2 \sqrt{10}-5}\) simplifies to \(\frac{2 \sqrt{10} + 5}{3}\) when the denominator is rationalized.

Step 1: Identify the Conjugate

Identify the conjugate of the denominator \(2 \sqrt{10}-5\). The conjugate is \(2 \sqrt{10}+5\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate. So we have, \(\frac{5}{2 \sqrt{10}-5} * \cfrac{2 \sqrt{10}+5}{2 \sqrt{10}+5}\). This does not change the value of the expression, it only changes its form.

Step 3: Simplify the Expression

In the numerator, distribute \(5 * (2 \sqrt{10}+5) = 10 \sqrt{10}+ 25.\) In the denominator, multiply using the difference of squares formula to get \(4*10 - 25 = 40 - 25 = 15.\) The expression now looks like \(\frac{10 \sqrt{10}+ 25}{15}\).

Step 4: Divide through by the Common Factor

The numerator and denominator have a common factor of \(5\). Divide through by the common factor to simplify the expression, giving the final answer: \(\frac{2 \sqrt{10} + 5}{3}\).

Key Concepts

Conjugate

When faced with a denominator containing a radical expression, using the conjugate is a powerful method for rationalizing it. The conjugate of a binomial expression like \(a \sqrt{b} - c\) is simply \(a \sqrt{b} + c\). This involves switching the sign in the middle of the two terms. It is an essential concept because multiplying by the conjugate results in a "difference of squares" pattern. This pattern helps in eliminating the square roots from the denominator, thereby rationalizing it.

• For expression \(2 \sqrt{10} - 5\), the conjugate is \(2 \sqrt{10} + 5\).
• Using the conjugate does not change the value of the expression, it only changes its form.

By multiplying both numerator and denominator by the conjugate, you will notice that the denominator simplifies into a rational number. This makes expressions easier to understand, particularly in further mathematical operations.

Difference of Squares

The difference of squares is a special pattern in algebra that is instrumental when dealing with conjugates. The pattern follows the formula: \((a - b)(a + b) = a^2 - b^2\). This remarkable pattern allows for quick simplification of expressions, especially when removing radicals from denominators.

• In our case, when multiplying \((2 \sqrt{10} - 5)(2 \sqrt{10} + 5)\), apply the difference of squares formula.
• You compute \( (2 \sqrt{10})^2 - 5^2 \) which results in \(4*10 - 25 = 40 - 25\).

The goal here is to achieve a simplified result of the multiplication where the radicals vanish, leaving a simple numerical expression for the denominator. This familiar pattern is what ultimately helps in lowering the complexity of such expressions.

Simplifying Expressions

Simplifying expressions is all about making them less complex, often by reducing them to their simplest form. After rationalizing the denominator, you generally face an expression that can still be made easier to work with.

• Once you have \(\frac{10 \sqrt{10}+ 25}{15}\), look for common factors.
• In this example, the greatest common divisor (GCD) between the terms in the numerator and 15 in the denominator is 5.
• By dividing top and bottom by the GCD: \(\frac{10 \sqrt{10} + 25}{15} \to \frac{2 \sqrt{10} + 5}{3}\).

Breaking down expressions into their simplest forms not only makes calculations easier but also helps in analyzing and understanding mathematical relations much more intuitively. It is an essential skill in algebra and forms the basis for further problem-solving in mathematics.