Problem 382 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Simplified form: \frac{ \sqrt{5y} + \sqrt{35} }{ y - 7 }

Step 1 - Identify the Conjugate

To rationalize the denominator, identify the conjugate of the denominator. The conjugate of \( \sqrt{y} - \sqrt{7} \ \) is \( \sqrt{y} + \sqrt{7} \ \).

Step 2 - Multiply by the Conjugate

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

Step 3 - Simplify the Denominator

Simplify the denominator using the difference of squares formula: \(( \sqrt{y} )^2 - ( \sqrt{7} )^2 = y - 7\). Thus, the denominator becomes: \[ ( \sqrt{y} )^2 - ( \sqrt{7} )^2 = y - 7 \]

Step 4 - Expand the Numerator

Expand the numerator using the distributive property: \[ \sqrt{5} \times ( \sqrt{y} + \sqrt{7} ) = \sqrt{5y} + \sqrt{35} \]

Step 5 - Combine and Simplify

Combining the results, the final expression is: \[ \frac{ \sqrt{5y} + \sqrt{35} }{ y - 7 } \]

Key Concepts

Conjugate

In order to rationalize the denominator, you need to use the conjugate of the denominator. The conjugate of a binomial expression of the form \( a - b \) is \( a + b \). In our exercise, the denominator is \( \sqrt{y} - \sqrt{7} \). Therefore, its conjugate would be \( \sqrt{y} + \sqrt{7} \). By multiplying both the numerator and the denominator by the conjugate, we essentially aim to eliminate the radicals from the denominator.
This helps in simplifying the expression and making it rational.

Difference of squares

The difference of squares formula states that \[ (a - b)(a + b) = a^2 - b^2 \]. This is a powerful tool when rationalizing denominators involving radicals. For the given exercise, the denominator is \( \sqrt{y} - \sqrt{7} \) and it is being multiplied by its conjugate \( \sqrt{y} + \sqrt{7} \).
Applying the difference of squares formula here gives us:
\( ( \sqrt{y} - \sqrt{7} )( \sqrt{y} + \sqrt{7} ) = ( \sqrt{y} )^2 - ( \sqrt{7} )^2 \).
This simplifies to \( y - 7 \). By using this formula, the radicals in the denominator are effectively removed.

Distributive property

The distributive property allows multiplication to be distributed over addition or subtraction. This is critically applied in the numerator of our exercise. After multiplying the numerator by the conjugate of the denominator, you get \( \sqrt{5} \cdot (\sqrt{y} + \sqrt{7} ) \).
Using the distributive property, this expands to: \( \sqrt{5} \cdot \sqrt{y} + \sqrt{5} \cdot \sqrt{7} \).
This results in: \( \sqrt{5y} + \sqrt{35} \). Applying the distributive property ensures that each term is multiplied appropriately, which is essential for correct simplification.

Radicals

Radicals involve the root of a number or expression. In the context of our exercise, we are dealing with square roots. When rationalizing the denominator with radicals, the goal is to eliminate these roots from the denominator to achieve a rational number.
Squaring radical expressions, as seen in steps when applying the difference of squares, often also simplifies the underlying expressions.

• For instance, \( (\sqrt{y})^2 = y \).
• Likewise, \( (\sqrt{7})^2 = 7 \).

Understanding how to manipulate radicals is crucial for solving similar problems and achieving the final simplified form.