Problem 28 - Algebra 1

Question

Simplify the expression. $$8 \sqrt{\frac{13}{9}}$$

Step-by-Step Solution

Answer

\(\frac{8\sqrt{13}}{3}\)

Step 1: Break Down the Square Root

To simplify this square root, one needs to take advantage of the fact that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). So, \(\sqrt{\frac{13}{9}}\) turns into \(\frac{\sqrt{13}}{\sqrt{9}}\)

Step 2: Simplify the Denominator

The denominator of the square root \(\sqrt{9}\) is a perfect square which simplifies to 3. The expression then becomes \(\frac{\sqrt{13}}{3}\)

Step 3: Multiply by the Coefficient

Now, multiply this by the coefficient 8 in the original problem. So, final answer is \(8 \times \frac{\sqrt{13}}{3}\)

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. Simplifying radical expressions allows for easier manipulation and understanding of algebraic equations. Take the expression \(8 \sqrt{\frac{13}{9}}\). This involves a square root, identified by the radical symbol \(\sqrt{}\), and numbers both inside and outside the root. The goal is to simplify this expression into its most reduced form.

To do this, one property we use is that the square root of a fraction can be expressed as the fraction of the square roots of the numerator and the denominator, or \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Applying this to our expression, it breaks down into \(8 \times \frac{\sqrt{13}}{\sqrt{9}}\) by treating the numerator and the denominator separately.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, and so on. Recognizing perfect squares is an essential skill when simplifying square roots because the square root of a perfect square is always an integer, which simplifies calculations and reduces complexity in an expression.

In our working example, we encounter the square root of 9, which is a perfect square (since \(3 \times 3 = 9\)). Therefore, \(\sqrt{9}\) simplifies directly to 3. This simplification is a crucial step because it makes the expression much more manageable as we then have \(8 \times \frac{\sqrt{13}}{3}\), with the denominator now rendered as a whole number rather than a radical.

Rationalizing Denominators

Rationalizing denominators is the process of eliminating radicals from the bottom (denominator) of fraction expressions. It often involves multiplying both the numerator and the denominator by a suitable expression that will clear the radical. However, when the denominator is already a perfect square, the process is greatly simplified because the square root of a perfect square is an integer.

In the context of the example, when the denominator under the radical sign becomes a perfect square, as with \(\sqrt{9}\), rationalization is straightforward. It simply involves applying the square root to the perfect square to obtain a rational number, which in our case is 3. No further action is required to 'rationalize' the denominator, and we seamlessly move onto using it in the final multiplication step to achieve the simplified expression.