Problem 14 - Elementary Algebra

Question

Simplify each square root. $$ \sqrt{\frac{a^{3}}{6}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sqrt{\frac{a^{3}}{6}}$$. Answer: $$\frac{a\sqrt{a}}{\sqrt{6}}$$

Step 1: Rewrite the expression using the square root property for fractions

Recall that the square root of a fraction can be written as the quotient of square roots: $$ \sqrt{\frac{a^{3}}{6}} = \frac{\sqrt{a^{3}}}{\sqrt{6}} $$

Step 2: Determine if there are any perfect squares within the expression

Check for any perfect squares in the numerator and the denominator: - \(a^3\) is not a perfect square because it has an odd exponent. - The number 6 is not a perfect square, but it can be factored as 2*3, and neither 2 nor 3 are perfect squares.

Step 3: Simplify the expression by taking the square root of any perfect squares

There are no perfect squares inside the square roots, but we can rewrite the numerator in terms of square roots: $$ \frac{\sqrt{a^{3}}}{\sqrt{6}} = \frac{a\sqrt{a}}{\sqrt{6}} $$ The numerator can be written as \(a\sqrt{a}\) since \(\sqrt{a^3} = a\sqrt{a}\).

Step 4: Combine the simplified terms to get the final answer

There are no more terms to simplify, so the final answer is: $$ \frac{a\sqrt{a}}{\sqrt{6}} $$

Key Concepts

Understanding the Square Root Property

The square root property is a fundamental tool that allows us to simplify square roots of fractions. When you encounter a square root that covers a fraction, remember the property:

• The square root of a fraction is the same as the fraction of square roots.

This means \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \] This property is especially useful because it separates the numerator and the denominator, allowing you to deal with each part individually. Applying this to the expression \(\sqrt{\frac{a^3}{6}}\), we split it into \(\frac{\sqrt{a^3}}{\sqrt{6}}\). This lays the groundwork for further simplification, as it helps uncover potential perfect squares or further simplifications within the simplified components.

Identifying Perfect Squares

Perfect squares are numbers or expressions that are the square of another number or expression. Recognizing them is crucial for simplifying square roots. If an expression is a perfect square, its square root is a whole number or a polynomial. Understanding this can make simplification much more straightforward.

• Perfect squares include numbers like 1, 4, 9, 16, and so on.
• Algebraic expressions like \(x^2\), \(4y^2\), or \((3x)^2\) are also perfect squares.

In our example, \(a^3\) is not a perfect square because its exponent is odd. However, we can express it as \(a^2 \cdot a\) so that \(a^2\), being a perfect square, simplifies to \(a\). The number 6, when broken into its factors 2 and 3, contains no perfect squares, making them irreducible within the square root.

Simplifying Expressions Effectively

Simplifying expressions involves breaking them down to their simplest form. The primary goal is to make the expression easier to understand or solve. It often involves using identities or properties like the square root property. To simplify \(\frac{\sqrt{a^3}}{\sqrt{6}}\), identify and separate any perfect squares. As we observed earlier, we can express \(\sqrt{a^3}\) as \(a\sqrt{a}\) by extracting the square \(\sqrt{a^2}\) as \(a\). The expression then becomes:\[\frac{a\sqrt{a}}{\sqrt{6}}\]There are no further reductions possible, as we can't simplify \(\sqrt{6}\) further. Always ensure to check for perfect squares and simplified real-number radicals to see if you can simplify further. Practice and familiarity with these steps will make simplifying expressions quicker and more intuitive.