Problem 25 - Algebra 1

Question

Simplify the expression. $$18 \sqrt{\frac{5}{81}}$$

Step-by-Step Solution

Answer

The simplified expression is \(6 \sqrt{5}\).

Step 1: Simplify Inside Square Root

We start by simplifying the fraction inside the square root by finding the square root of the denominator. The square root of 81 is 9. Hence, the expression inside the square root becomes \(\frac{5}{9}\). Therefore our expression is now \(18 \sqrt{\frac{5}{9}}\).

Step 2: Distribute the Square Root

The square root of a fraction can be written as the square root of the numerator over the square root of the denominator according to the property of radicals \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). Hence, our expression becomes \(18 \frac{\sqrt{5}}{3}\).

Step 3: Simplify the Expression

Now we simplify the whole expression. Multiply 18 by \(\frac{1}{3}\) to simplify the expression which equals 6. Hence, the final simplified expression is \(6 \sqrt{5}\).

Key Concepts

Square Roots

When we talk about square roots, we're referring to a mathematical operation where you find a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9.
Square roots are often represented with the radical symbol (√). For instance, the square root of a number ‘x’ is written as \( \sqrt{x} \). It's important to remember that numbers can have both positive and negative square roots since both (\(3 \times 3\)) and (\(-3 \times -3\)) will yield 9. However, when using the square root symbol, we generally refer to the principal (or positive) square root.
In the original expression, the inside of the square root contains a fraction \( \sqrt{\frac{5}{81}} \). We simplify this by finding square roots of both the numerator and the denominator separately.

Fraction Simplification

Fractions can be tricky, but with a little knowledge, they become much easier to handle. Simplifying fractions involves altering the fraction so that its numerator and denominator are as small as possible while still maintaining the same value.
In general, to simplify a fraction, you should:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

In the context of square roots, it's crucial to simplify any fraction inside a radical by separating the numerator and the denominator under individual square roots. For the exercise given, \( \sqrt{\frac{5}{81}} \) simplifies inside the root to \( \frac{\sqrt{5}}{\sqrt{81}} \) because each part of the fraction is handled separately, leading to \( \frac{\sqrt{5}}{9} \). This step is vital because it sets the stage for combining other values outside the radical.

Properties of Radicals

Radicals often have specific properties that can help simplify calculations, especially when dealing with fractions or complex algebraic expressions.
One of the critical properties is that a square root of a fraction is equal to dividing the square root of the numerator by the square root of the denominator: \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \] This property is particularly useful in situations where both the numerator and the denominator are perfect squares or can be simplified further individually, as seen in the expression \( \sqrt{\frac{5}{81}} \). When applying this property, as in the provided exercise, the calculation becomes simpler and avoids unnecessary complexity.
Successful application of these properties allows the simplification of radicals across fractions, making expressions much easier to understand and manipulate, as seen with the final product, \( 18 \times \frac{\sqrt{5}}{3} \), which elegantly reduces down to \( 6 \sqrt{5} \).