Problem 33 - Algebra 1

Question

Simplify the expression. $$8 \sqrt{\frac{20}{4}}$$

Step-by-Step Solution

Answer

The simplified expression is \(8 * \sqrt{5}\)

Step 1: Simplify the Fraction

Start by simplifying the fraction \(\frac{20}{4}\) under the square root. \(\frac{20}{4} = 5\)

Step 2: Calculate the Square Root

After simplifying the fraction, now calculate the square root of the resultant number \( \sqrt{5} \)

Step 3: Multiply by the Coefficient

Multiply the result from the square root step by the coefficient 8. The final step is \(8 * \sqrt{5}\) which equals \(8*\sqrt{5}\)

Key Concepts

Simplify Expressions

Understanding how to simplify expressions is a foundational skill in algebra. This involves reducing expressions to their simplest form, making them easier to work with. When simplifying expressions, especially those that contain square roots, we first look for any possible simplification within the square root, and then we apply the square root to those simplified values. Simplifying can involve factoring, combining like terms, or simplifying fractions. This leads us more smoothly into our calculations and results in a more comprehensible expression. In the given exercise, the expression starts by simplifying what's inside the square root, which sets the stage for further steps.

Fraction Simplification

In any mathematical problem involving fractions, fraction simplification is a crucial initial step. Simplifying a fraction means to reduce it to its lowest terms, where the numerator and the denominator have no common factors other than 1. To do so, we divide both the numerator and the denominator by their greatest common divisor. For example, with the fraction \( \frac{20}{4} \) from the exercise, since both 20 and 4 are divisible by 4, we simplify this fraction to \( \frac{5}{1} \) or simply 5. This simplified fraction then makes subsequent calculations much more straightforward.

Square Root Calculation

The process of square root calculation is often a source of difficulty. The square root of a number is a value that, when multiplied by itself, will result in the original number. For fractions, the square root can be applied to both the numerator and the denominator separately if they are both perfect squares. However, when dealing with numbers that are not perfect squares, such as 5 in our example, it's vital to leave the square root in its radical form (like \( \sqrt{5} \) in this case) rather than converting it to a decimal, which can lead to loss of precision in more complex calculations.

Algebraic Coefficients

An understanding of algebraic coefficients is important when working with algebraic expressions. The coefficient is a constant multiplier of the variable or variables in the term. In the case of our exercise, the number 8 is the coefficient that is multiplied by the square root of 5. When you have a coefficient outside of a radical, such as 8 in \(8 * \sqrt{5}\), it is separated from the operation of the radical. You multiply the coefficient by the square root after the value under the root has been simplified, ensuring the expression stays in its most accurate and simplest radical form.