Problem 88 - Intermediate Algebra

Question

Rationalize each denominator and simplify if possible. $$\frac{10}{\sqrt{5}}$$

Step-by-Step Solution

Answer

\( \frac{10}{\sqrt{5}} = 2\sqrt{5} \).

Step 1: Identify the Problem

We have a fraction \( \frac{10}{\sqrt{5}} \) and the problem asks us to rationalize the denominator, which means eliminating any square roots from the denominator.

Step 2: Multiply by a Conjugate

To eliminate the square root from the denominator, we will multiply both the numerator and the denominator of \( \frac{10}{\sqrt{5}} \) by \( \sqrt{5} \), which is the conjugate of \( \sqrt{5} \).

Step 3: Apply Multiplication

Multiply the numerator and the denominator: \( \frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10 \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} \). This simplifies to \( \frac{10 \sqrt{5}}{5} \), because \( \sqrt{5} \cdot \sqrt{5} = 5 \).

Step 4: Simplify the Fraction

Simplify \( \frac{10 \sqrt{5}}{5} \) by dividing both the numerator and the denominator by 5. This gives \( 2 \sqrt{5} \), as \( \frac{10}{5} = 2 \).

Step 5: Confirm Result

After rationalizing and simplifying, the expression \( \frac{10}{\sqrt{5}} \) becomes \( 2 \sqrt{5} \). We have successfully rationalized the denominator.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial skill in algebra. It involves reducing the numerator and denominator to the lowest terms without changing the value.

• Start by finding the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both without leaving a remainder.
• Divide both the numerator and the denominator by their GCD.

For example, in the fraction \( \frac{10 \sqrt{5}}{5} \), both the numerator (10) and the denominator (5) can be divided by 5. This simplifies to \( 2 \sqrt{5} \), as \( \frac{10}{5} = 2 \).
Simplification of fractions makes them easier to understand and work with, especially in further mathematical calculations.

Square Roots

Square roots are numbers that, when multiplied by themselves, give the original number. The symbol for a square root is \( \sqrt{} \).

• The square root of a perfect square, like 25, is an integer (e.g., \( \sqrt{25} = 5 \)).
• Non-perfect squares, like 5, result in irrational numbers (e.g., \( \sqrt{5} \)), which cannot be neatly expressed as a fraction.

When dealing with fractions, square roots in the denominator can complicate things. This is why rationalizing denominators is often necessary. By multiplying the numerator and the denominator by the same square root, as shown in the example \( \frac{10}{\sqrt{5}} \), you eliminate the square root in the denominator.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like x or y), and arithmetic operations.

• They can represent constants, equations, or inequalities and can be as simple or complex as needed.
• Simplifying these expressions often involves combining like terms or factoring.

In the case of rationalizing \( \frac{10}{\sqrt{5}} \), the expression initially involves the division of a decimal by a square root. Transforming this expression into \( 10 \sqrt{5} \) over 5 and then simplifying, brings it into a simpler algebraic expression \( 2 \sqrt{5} \).
This process of managing algebraic expressions is fundamental for solving equations or optimizing functions in mathematics.