91Ó°ÊÓ

When working with square roots in algebra, one handy rule to remember is the Product Property of Square Roots. This property states that the square root of a product is equal to the product of the square roots. That means, for any positive numbers ‘a’ and ‘b’, we can write:
ewline ewline \[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \] ewline ewline Using this property, we can combine two separate square roots into one. For the given problem, applying the product property \sqrt{3 b^{3}} \cdot \sqrt{6 b^{5}} means simply creating a single square root:ewline ewline \[ \sqrt{3 b^{3}} \cdot \sqrt{6 b^{5}} = \sqrt{(3 b^{3}) \cdot (6 b^{5})} \] This makes it easier to tackle the simplification process as we move forward with our calculations.

Multiplying exponents can seem tricky, but with the right approach, it becomes straightforward. There are a few rules you need to know. ewline ewline \begin{itemize} \item When multiplying like bases, you add the exponents: \[ x^a \cdot x^b = x^{a+b} \] \item Always multiply the numerical coefficients first. In our case: \[ 3 \cdot 6 = 18 \] \end{itemize} ewline ewline So, in our original problem, multiply the constants and add the exponents for the variable ‘b’:ewline ewline \[ 3 b^{3} \cdot 6 b^{5} = 18 b^{3+5} = 18 b^{8} \] Now, you have transformed the product of two expressions inside the square root.

Simplifying square roots involves breaking the expression down into parts that make simplification easier. When we reached \sqrt{18 b^{8}}, we needed to rewrite the expression to simplify it. Look for perfect squares inside the square root. ewline ewline \begin{itemize} \item 18 can be broken into factors: \[ 18 = 9 \cdot 2 \] where 9 is a perfect square. ewline ewline \item For the variable part, rewrite it as squares: \[ b^{8} = (b^{4})^2 \] \end{itemize} ewline ewline So, the expression inside the square root becomes: \[ \sqrt{18 b^{8}} = \sqrt{9 \cdot 2 \cdot (b^4)^2} \] ewline ewline Now, apply the square root to the perfect squares: \[ \sqrt{9 \cdot 2 \cdot (b^4)^2} = 3 b^{4} \sqrt{2} \] By breaking it down, we have simplified the expression fully.