Problem 318 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{\sqrt{50}}{10} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{2}}{2} \)

Step 1: Simplify the square root

Start by simplifying the square root in the numerator. Notice that 50 can be factored into 25 and 2. Since 25 is a perfect square, it can be simplified as follows:\[ \ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]

Step 2: Substitute back into the fraction

Replace \( \sqrt{50} \) in the original fraction with the simplified form \( 5\sqrt{2} \):\[ \ \frac{\sqrt{50}}{10} = \frac{5\sqrt{2}}{10} \]

Step 3: Simplify the fraction

To simplify the fraction \( \frac{5\sqrt{2}}{10} \), divide both the numerator and the denominator by 5:\[ \ \frac{5\sqrt{2}}{10} = \frac{5}{10} \cdot \sqrt{2} = \frac{1}{2} \cdot \sqrt{2} = \frac{\sqrt{2}}{2} \]

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing the numerator and the denominator to their smallest possible values. In the case of a fraction like \( \frac{5\sqrt{2}}{10} \), you should find the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD is 5.

Divide both parts by 5 to get \( \frac{5\sqrt{2}}{10} = \frac{1}{2} \sqrt{2} \). This makes the fraction less complicated and easier to work with.

• Look for a common factor in the numerator and the denominator.
• Divide both by this common factor.
• The fraction is now simplified.

Breaking down a fraction into simpler terms can greatly aid in further calculations.

Perfect Squares

Understanding perfect squares is foundational when simplifying square roots. A perfect square is a number that is the square of an integer. For example, 25 is a perfect square because it is \(5^2 = 25\).

When simplifying \( \sqrt{50} \), notice that 50 can be written as 25 times 2. Since 25 is a perfect square, it simplifies to 5: \( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \).

• Identify any perfect squares within the number under the square root.
• Simplify the square root of the perfect square.

Understanding perfect squares helps streamline the process of dealing with square roots.

Radicals

Radicals often appear in mathematics and understanding them is crucial for simplifying expressions. A radical is an expression that includes a square root, cube root, or other roots. In this case, we're dealing with square roots.

To simplify a radical like \( \sqrt{50} \), you need to factor the number into its constituent parts. Look for perfect squares among these factors for simplification and reduce it, resulting in \( 5\sqrt{2} \).

• Factor the number under the radical.
• Look for any perfect squares.
• Simplify the root and any remaining parts.

Mastering how to handle radicals can make many complex math problems much easier to solve.