Simplify square root of 20a^4b^5
The given problem "Simplify square root of 20a^4b^5" is asking for the simplification of the radical expression involving the square root of a product of a number and variables raised to various powers. Specifically, it requires you to find the simplest radical form of the square root of the product, which includes the number 20 and the variables 'a' raised to the fourth power and 'b' raised to the fifth power. The simplification will likely involve factoring the number and the powers of the variables into perfect squares, since the square root operation is applied to them.
$\sqrt{20 a^{4} b^{5}}$
Answer
Solution:
Step 1: Express the given expression in a factorized form.
Rewrite the expression $\sqrt{20a^4b^5}$ by identifying perfect squares and other factors.
Step 1.1
Extract the factor of $4$ from $20$, resulting in $\sqrt{4 \cdot 5a^4b^5}$.
Step 1.2
Express $4$ as a square of $2$, giving us $\sqrt{2^2 \cdot 5a^4b^5}$.
Step 1.3
Represent $a^4$ as $(a^2)^2$, leading to $\sqrt{2^2 \cdot 5(a^2)^2b^5}$.
Step 1.4
Separate out $b^4$ from $b^5$, yielding $\sqrt{2^2 \cdot 5(a^2)^2(b^4 \cdot b)}$.
Step 1.5
Write $b^4$ as $(b^2)^2$, resulting in $\sqrt{2^2 \cdot 5(a^2)^2((b^2)^2 \cdot b)}$.
Step 1.6
Rearrange to place $5$ at the end, obtaining $\sqrt{2^2(a^2)^2(b^2)^2 \cdot 5b}$.
Step 1.7
Combine the squares into a single term, getting $\sqrt{(2a^2b^2)^2 \cdot 5b}$.
Step 1.8
Enclose the terms properly with parentheses, resulting in $\sqrt{((2a^2b^2)^2 \cdot (5b))}$.
Step 2: Simplify the square root.
Extract terms that are perfect squares from under the square root to get $2a^2b^2\sqrt{5b}$.
Knowledge Notes:
To simplify a square root involving variables and constants, we follow these steps:
Factorization: Break down the expression inside the square root into factors, specifically looking for perfect squares and other terms.
Perfect Squares: Recognize that the square root of a perfect square is simply the base of the square. For example, $\sqrt{a^2} = a$ and $\sqrt{2^2} = 2$.
Variable Exponents: For variables under a square root, even exponents indicate that the variable can be taken out of the square root without any radical left. For example, $\sqrt{a^4} = a^2$.
Combining Like Terms: Group the perfect square factors together to simplify the expression under the square root.
Simplification: Extract the perfect square factors from under the square root, while leaving the other factors inside. This is done because the square root of a product is equal to the product of the square roots of the individual factors, provided all quantities are non-negative.
In this problem, we applied these principles to simplify $\sqrt{20a^4b^5}$, ultimately extracting $2a^2b^2$ as the perfect square and leaving $\sqrt{5b}$ as the remaining factor under the square root.
