Problem 49 - Algebra 1

Question

Simplify the expression. $$\frac{-2 \sqrt{20}}{\sqrt{100}}$$

Step-by-Step Solution

Answer

The simplified expression is -2√5/5.

Step 1: Find the square roots

Firstly, simplify the square roots. The square root of 20 can be simplified by searching for possible perfect squares that could yield whole numbers. Here, 4 * 5 equals 20. Thus, \( \sqrt{20} \) equals 2 \( \sqrt{5} \). Additionally, \( \sqrt{100} \) equals 10. Hence, the expression can be rewritten as: \(\frac{-2 \times 2\sqrt{5} }{10}\).

Step 2: Divide the expression

Next, divide the numerator and the denominator by 2. Remembering that square root of 5 in the numerator is multiplied by 4, not 2, because there are two of them: -2 and 2 coming from \( \sqrt{20} \). Thus, the expression simplifies to \( \frac{-4\sqrt{5} }{10}\).

Step 3: Simplify the result

Finally, simplify the fraction, by dividing the numerator and the denominator by common factor, which is 2. Hence, the simplified result is \( \frac{-4\sqrt{5} }{10} \) equals \( \frac{-2\sqrt{5} }{5}\).

Key Concepts

Square Roots

Square roots are mathematical operations that "undo" the action of squaring. When you square a number, you multiply it by itself. The square root of this number is the value that, when multiplied by itself, yields the original number. For example, the square root of 4 is 2 because 2 multiplied by 2 equals 4.
To simplify square roots, look for perfect square factors, such as 4, 9, 16, which have whole numbers as square roots. In the expression \(\sqrt{20}\), recognize that 4, being a perfect square, is a factor of 20. Hence, \(\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}\). This makes calculations easier and can help simplify expressions.

Rationalizing Denominators

Rationalizing the denominator means ensuring the denominator of a fraction is a rational number (i.e., no square roots or irrational numbers remain in it).
In our example, \(\frac{-2 \sqrt{20}}{\sqrt{100}}\), the denominator \(\sqrt{100}\) simplifies to 10, a rational number. Therefore, in this case, rationalizing was straightforward as the square root was already a perfect square.
This process is crucial for situations where the denominator contains square roots. By rationalizing, we make expressions easier to handle, especially when we need further mathematical operations.

Algebraic Fractions

Algebraic fractions are expressions that contain polynomials or irrational numbers in the numerator, denominator, or both. Managing them involves similar techniques as for numeric fractions, such as simplifying by finding common factors.
In the expression \(\frac{-2\sqrt{20}}{\sqrt{100}}\), we rewrite the numerator as \(-4\sqrt{5}\) through simplification, using the product representation learned in square roots. By dividing the top and bottom by their greatest common divisor (GCD), here being 2, we further simplify to obtain \(\frac{-2\sqrt{5}}{5}\).
Understanding how to manipulate algebraic fractions is essential because these concepts recur frequently in algebra and higher math, improving problem-solving efficiency.