Simplify square root of 48t^9w^10
The question asks for the simplification of the square root of an algebraic expression. Specifically, it's asking you to simplify the square root of the product of 48 and two variables raised to powers, namely 't' raised to the 9th power and 'w' raised to the 10th power. The goal is to simplify this expression using the properties of square roots and exponents to find an equivalent expression that is simpler and has no radical sign, if possible, or has the radical sign with the simplest possible radicand.
$\sqrt{48 t^{9} w^{10}}$
Answer
Solution:
Step 1: Decompose the expression
Break down $48 t^{9} w^{10}$ into its prime factors and perfect squares.
Step 1.1
Extract the square factor from $48$. $\sqrt{16 \cdot 3 t^{9} w^{10}}$
Step 1.2
Express $16$ as a square of $4$. $\sqrt{4^2 \cdot 3 t^{9} w^{10}}$
Step 1.3
Separate the perfect square $t^8$ from $t^9$. $\sqrt{4^2 \cdot 3 (t^8 t) w^{10}}$
Step 1.4
Represent $t^8$ as $(t^4)^2$. $\sqrt{4^2 \cdot 3 ((t^4)^2 t) w^{10}}$
Step 1.5
Express $w^{10}$ as $(w^5)^2$. $\sqrt{4^2 \cdot 3 ((t^4)^2 t) (w^5)^2}$
Step 1.6
Rearrange $t$. $\sqrt{4^2 \cdot 3 ((t^4)^2) (w^5)^2 t}$
Step 1.7
Rearrange $3$. $\sqrt{4^2 ((t^4)^2) (w^5)^2 \cdot 3 t}$
Step 1.8
Combine the squares into a single term. $\sqrt{(4 t^4 w^5)^2 \cdot 3 t}$
Step 1.9
Enclose the expression in parentheses. $\sqrt{((4 t^4 w^5)^2 \cdot (3 t))}$
Step 2: Simplify the radical
Extract the square terms from the square root. $4 t^4 w^5 \sqrt{3 t}$
Knowledge Notes:
To simplify a square root involving variables and constants, we follow these steps:
Prime Factorization: Break down the constants into their prime factors to identify any perfect squares.
Identify Perfect Squares: Look for perfect squares in the variables' exponents, as these can be taken out of the square root.
Rewrite as Squares: Express the constants and variables that are perfect squares in the form of $(\text{base})^2$ to prepare them to be taken out of the square root.
Simplify Inside the Radical: Rearrange the terms inside the square root to separate perfect squares from non-perfect squares.
Extract Perfect Squares: Take the square root of the perfect squares, which can be done by removing the square root symbol and halving the exponents of the variables.
Simplify the Expression: Combine the terms outside the square root and leave the rest inside the square root.
In this problem, we used the fact that $\sqrt{a^2} = a$ for any non-negative real number $a$ and that $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ to separate and simplify the terms under the radical. The goal is to express the original expression in its simplest radical form, where no perfect square factors remain inside the square root.
