Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rationalize the denominator. $$\frac{10}{\sqrt{14}-2}$$

Step-by-Step Solution

Answer

The rationalized form of the fraction \(\frac{10}{\sqrt{14}-2}\) is \(\sqrt{14} + 2\).

Step 1: Identify the Conjugate

The conjugate of a binomial expression \(\sqrt{14}-2\) is \(\sqrt{14}+2\). Conjugates are useful for rationalizing denominators because they eliminate radicals when multiplied.

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator. That is,\[\frac{10}{\sqrt{14}-2} \times \frac{\sqrt{14}+2}{\sqrt{14}+2}\]

Step 3: Simplify the Expression

When multiplying out the expressions, note that the denominator will become a difference of squares which simplifies to:\[(\sqrt{14})^2 - (2)^2 = 14 - 4 = 10\]So the denominator simplifies to 10. In the numerator, distribute the 10 to get \[10 \times \sqrt{14}+10 \times 2 = 10\sqrt{14} + 20\]So the numerator simplifies to \(10\sqrt{14} + 20\).

Step 4: Final Solution

Therefore, the rationalized form of the fraction is\[\frac{10\sqrt{14} + 20}{10}\]

Step 5: Simplification

Simplify your answer by dividing both terms in the numerator by 10 to get\[\sqrt{14} + 2\]

Key Concepts

Conjugate of Binomial Expression

When dealing with binomials involving square roots, a powerful tool in mathematics is the conjugate. The conjugate of a binomial expression such as \( \sqrt{14} - 2 \) is derived simply by changing the sign between its terms. Therefore, the conjugate of \( \sqrt{14} - 2 \) is \( \sqrt{14} + 2 \). This concept is crucial in rationalizing denominators because the product of a binomial and its conjugate results in a simplified expression.

The use of conjugates is particularly beneficial in eliminating radicals from denominators. This occurs because when a binomial is multiplied by its conjugate, the result is a difference of squares, leading to simplified whole numbers.

Difference of Squares

The term 'difference of squares' refers to a special algebraic identity. This identity states that for any two numbers \(a\) and \(b\), the expression \((a - b)(a + b) = a^2 - b^2\). This property is what makes conjugates so valuable in rationalizing denominators.

In our exercise, multiplying the conjugate yields \((\sqrt{14})^2 - (2)^2\), which simplifies to \(14 - 4 = 10\). This calculation eradicates the square root in the denominator, producing a much simpler integer value.

The difference of squares helps to turn complex expressions into manageable forms, aiding in further simplification.

Multiplying Radicals

When multiplying radicals, the rules of exponentiation apply. Specifically, the product of two radicals with the same index equals a single radical with the product of the original radicands. For example, \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \).

In our example problem, the numerator \(10\) is multiplied by the conjugate \(\sqrt{14} + 2\). As a result, each term of the binomial is distributed by the multiplier resulting in:

• \(10 \cdot \sqrt{14} = 10\sqrt{14}\)
• \(10 \cdot 2 = 20\)

These multiplications maintain the radical in its neat form while applying the constant factor.

Simplifying Expressions

Simplifying expressions is the process of reducing fractions or removing complexity from a term to render it in its most straightforward form. Once the expression \(\frac{10\sqrt{14} + 20}{10}\) was achieved from multiplying by the conjugate, the next logical step was simpilfying this result.

This involved dividing each term in the numerator by the common factor in the denominator, which was \(10\):

• \(\frac{10\sqrt{14}}{10} = \sqrt{14}\)
• \(\frac{20}{10} = 2\)

Thus, the simplified form of the expression became \(\sqrt{14} + 2\). Simplifying expressions not only makes them easier to interpret but also aids in comparing mathematical relationships.