Simplify square root of 12a^8b^9

Solution:

Step 1: Express the given expression in a form that separates perfect squares from other factors.

Rewrite $\sqrt{12a^8b^9}$ by identifying perfect square factors and separating them from non-perfect square factors.

Step 1.1

Identify that $12$ can be factored into $4 \times 3$, where $4$ is a perfect square. Thus, we have $\sqrt{4 \cdot 3 a^8 b^9}$.

Step 1.2

Express $4$ as the square of $2$, which is $2^2$. This gives us $\sqrt{2^2 \cdot 3 a^8 b^9}$.

Step 1.3

Notice that $a^8$ is a perfect square since it can be written as $(a^4)^2$. We now have $\sqrt{2^2 \cdot 3 (a^4)^2 b^9}$.

Step 1.4

Similarly, factor $b^9$ as $b^8 \cdot b$, where $b^8$ is a perfect square. We get $\sqrt{2^2 \cdot 3 (a^4)^2 (b^8 b)}$.

Step 1.5

Rewrite $b^8$ as $(b^4)^2$, which is another perfect square. Our expression becomes $\sqrt{2^2 \cdot 3 (a^4)^2 ((b^4)^2 b)}$.

Step 1.6

Rearrange the terms to group the perfect squares together. We have $\sqrt{2^2 (a^4)^2 (b^4)^2 \cdot 3b}$.

Step 1.7

Combine the perfect squares under a single square root to form $(2 a^4 b^4)^2$. The expression is now $\sqrt{(2 a^4 b^4)^2 \cdot 3b}$.

Step 1.8

Ensure the expression is properly parenthesized. We have $\sqrt{((2 a^4 b^4)^2 \cdot (3b))}$.

Step 2: Simplify the square root by taking out the perfect square terms.

Extract the perfect square terms from under the square root, which gives us $2 a^4 b^4 \sqrt{3b}$.

Knowledge Notes:

To simplify a square root involving variables and coefficients, we follow these steps:

1. Factor the coefficient into its prime factors and identify any perfect squares.

2. Express each variable's exponent as a multiple of 2 if possible, to identify perfect square factors.

3. Rewrite the expression by separating the perfect square factors from the non-perfect square factors.

4. Take the square root of the perfect square factors by removing them from under the square root sign and halving their exponents.

5. Combine the terms outside the square root and leave the remaining terms that are not perfect squares under the square root.

In this problem, we used the fact that the square root of a perfect square is simply the base of the exponent. For example, $\sqrt{a^{2n}} = a^n$ for any non-negative integer $n$. We also used the property that the square root of a product is the product of the square roots, provided all quantities involved are non-negative. This allows us to separate terms under the square root for simplification.