Simplifying Complex Algebraic Expressions

When dealing with algebraic expressions, it's important to understand how to simplify them to make them easier to work with. In this article, we'll walk through the step-by-step process of simplifying the expression: \frac{(9a^{3}b^{4})^{1/2}}{15a^{2}b}.

Step 1: Simplify the numerator

The numerator of the expression is (9a^{3}b^{4})^{1/2}. Let's break this down:

• 9a^{3}b^{4} means we have 9 multiplied by a raised to the power of 3, and b raised to the power of 4.
• The exponent 1/2 represents a square root. So, we need to take the square root of 9a^{3}b^{4}.

To take the square root, we can use the rule: (x^n)^{1/m} = x^{n/m}. Applying this rule, we get: (9a^{3}b^{4})^{1/2} = 3a^{3/2}b^{2}

Step 2: Simplify the denominator

The denominator of the expression is 15a^{2}b. We can leave this as is, as it is already in its simplified form.

Step 3: Combine the numerator and denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final simplified expression: \frac{3a^{3/2}b^{2}}{15a^{2}b}

To further simplify this expression, we can use the rule of exponents: a^m/a^n = a^{m-n}. Applying this rule, we get: \frac{3a^{3/2}b^{2}}{15a^{2}b} = \frac{3a^{3/2-2}b^{2-1}}{15} = \frac{3a^{-1/2}b}{15}

Therefore, the simplified expression is \frac{3a^{-1/2}b}{15}.