Question

Simplify. $$\sqrt{\frac{3 a^{3}}{5 b}}$$

Short answer

\(\frac{a\sqrt{3a}}{\sqrt{15b}}\)

Step-by-step solution

Identify Perfect Squares

To simplify the given square root, \(\sqrt{\frac{3 a^{3}}{5 b}}\), identify any perfect squares in the numerator and the denominator. However, in this case, there are no perfect squares in the radicand (the expression inside the square root).

Factor Out Powers of 2

Since we are dealing with a square root, we can simplify by factoring out any powers of 2 from the variables. In the expression, we can factor out \(a^2\) from \(a^3\) as \(a^3 = a^2 \cdot a\).

Rewrite with Factored Terms

Rewrite the radicand by separating the factored powers of 2, giving us \(\sqrt{\frac{3\cdot a^2\cdot a}{5 b}}\).

Simplify Square Root

Take the square root of any perfect squares out of the radical. We have \(a^2\) which is a perfect square and its square root is \(a\). Thus, we have \(\sqrt{\frac{3\cdot a^2}{5 b}}\cdot\sqrt{a}\), which simplifies to \(\frac{a}{\sqrt{\frac{5 b}{3}}}\cdot\sqrt{a}\).

Rationalize the Radical

To rationalize the denominator of a fraction within a radical, multiply both the numerator and the denominator inside the radical by any terms needed to make a perfect square under the radical. In this case, we multiply both by 3 to get \(\sqrt{\frac{3\cdot a^2\cdot 3\cdot a}{5 b\cdot 3}}\), which simplifies to \(\frac{a\cdot\sqrt{3\cdot a}}{\sqrt{5 b\cdot 3}} = \frac{a\cdot\sqrt{3a}}{\sqrt{15b}}\).

Combine and Simplify

Combine the terms to obtain the simplest form and then simplify the square roots if possible. \(\frac{a\cdot\sqrt{3a}}{\sqrt{15b}}\) is already in its simplest form as there are no more perfect squares to take out and no further simplification can be done. So, the original expression simplifies to \(\frac{a\sqrt{3a}}{\sqrt{15b}}\).

Key concepts

Radicals Simplification

Understanding how to simplify radicals, or square roots, is a valuable skill in algebra. Simplifying radicals involves finding the prime factorization of the number inside the radical and then identifying pairs of identical factors, as we can take the square root of any pair of identical factors. For example, \( \sqrt{36} \) simplifies to 6 because 36 is a perfect square, being \( 6 \times 6 \). When simplifying the square root of an expression with variables, such as \( \sqrt{a^3} \), we look for the highest power of the variable that is a multiple of 2 as square roots involve pairs of factors. In this case, we would take \( a^2 \) out of the radical, leaving \( a \) inside the radical since \( a^3 = a^2 \times a \).

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. This process makes it easier to work with and understand the expression, and is also considered a formal requirement in mathematics. To rationalize, we multiply the numerator and denominator by a number that will create a perfect square under the radical in the denominator. As seen in the exercise, to rationalize the denominator of \( \sqrt{\frac{5b}{3}} \), we'd multiply by \( \sqrt{3} \) because \( 5b \times 3 \) is a multiple of a perfect square (in this case, 3). Thus, it becomes \( \sqrt{\frac{15b}{9}} \) or \( \sqrt{\frac{15b}{3^2}} \), which simplifies the radical since the denominator is now a perfect square.

Factoring Algebraic Expressions

Factoring algebraic expressions is a process by which we express a polynomial or other algebraic expression as a product of simpler factors. This technique is not only useful in simplifying expressions but also in solving equations and understanding polynomial functions. In the given exercise, we use factoring to simplify the variable portion of the radical. For instance, the algebraic expression \( a^3 \) is factored out as \( a^2 \times a \) to separate out the largest square factor, which in this case is \( a^2 \). Factoring is particularly useful in simplifying radicals because we are often looking to find and extract perfect squares to simplify the square root.