Problem 368 - Elementary Algebra

Question

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{5}}{\sqrt{n}-\sqrt{7}} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt{5n} + \sqrt{35}}{n - 7}\)

Step 1: Identify the Conjugate

The conjugate of the denominator \(\sqrt{n} - \sqrt{7}\) is \(\sqrt{n} + \sqrt{7}\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(\sqrt{n} + \sqrt{7}\). This gives: \[\frac{\sqrt{5}}{\sqrt{n} - \sqrt{7}} \times \frac{\sqrt{n} + \sqrt{7}}{\sqrt{n} + \sqrt{7}} = \frac{\sqrt{5}(\sqrt{n} + \sqrt{7})}{(\sqrt{n} - \sqrt{7})(\sqrt{n} + \sqrt{7})}\]

Step 3: Simplify the Denominator

Using the difference of squares formula, simplify the denominator: \((\sqrt{n})^2 - (\sqrt{7})^2 = n - 7\). Now the fraction looks like \[\frac{\sqrt{5}(\sqrt{n} + \sqrt{7})}{n - 7}\]

Step 4: Expand the Numerator

Expand the numerator: \[\sqrt{5} \cdot \sqrt{n} + \sqrt{5} \cdot \sqrt{7} = \sqrt{5n} + \sqrt{35}\]

Step 5: Combine the Results

Combine the results to get the simplified expression: \[\frac{\sqrt{5n} + \sqrt{35}}{n - 7}\]

Key Concepts

Conjugate in algebra

In algebra, the conjugate of a binomial expression involves changing the sign between two terms. For example, the conjugate of \(\backslash\backslash\backslashsqrt{a} - \backslash\backslash\backslashsqrt{b} \) is \(\backslash\backslash\backslashsqrt{a} + \backslash\backslash\backslashsqrt{b} \). This technique is essential for rationalizing denominators, as it allows us to eliminate irrational numbers from the denominator by leveraging the difference of squares formula.

By multiplying a fraction's numerator and denominator by the conjugate of the denominator, we can simplify complex expressions. The process ensures that the denominator becomes a whole number, making the overall expression more manageable and easier to understand.

Difference of squares

The difference of squares is a mathematical formula that states: \[ a^2 - b^2 = (a - b)(a + b). \] This identity helps us simplify expressions involving the product of a binomial and its conjugate.

For instance, in our exercise, the denominator was \(\backslash\backslash\backslashsqrt{n} - \backslash\backslash\backslashsqrt{7} \). When multiplied by its conjugate \(\backslash\backslash\backslashsqrt{n} + \backslash\backslash\backslashsqrt{7} \), the result is \(\backslash\backslash\backslashsqrt{n}^2 - \backslash\backslash\backslashsqrt{7}^2 \): \(\backslash\backslash\backslashsqrt{n}^2 - \backslash\backslash\backslashsqrt{7}^2 = n - 7 \).

This transformation simplifies our expression by converting the denominator into a simpler form, eventualizing less fractional complexity.

Simplification in algebra

Simplification in algebra means rewriting an expression in its most simplified form. This involves performing operations like combining like terms, reducing fractions, and using identities such as the difference of squares.

In the given exercise, we simplified by first rationalizing the denominator. We used the conjugate to transform the denominator into a simpler form: \((\backslash\backslash\backslashsqrt{n} - \backslash\backslash\backslashsqrt{7})(\backslash\backslash\backslashsqrt{n} + \backslash\backslash\backslashsqrt{7}) = n - 7 \).

We then simplified the numerator by expanding it: \(\backslash\backslash\backslashsqrt{5} \backslash\backslash\backslashcdot \backslash\backslash\backslashsqrt{n} + \backslash\backslash\backslashsqrt{5} \backslash\backslash\backslashcdot \backslash\backslash\backslashsqrt{7} = \backslash\backslash\backslashsqrt{5n} + \backslash\backslash\backslashsqrt{35} \).

Finally, we combined these results to obtain the rationalized expression: \[\frac{\backslash\backslash\backslashsqrt{5n} + \backslash\backslash\backslashsqrt{35}}{n - 7} \]