Simplify square root of 48p
The problem you're presented with is an algebraic expression simplification task. It involves taking the square root of a term, specifically "48p," and rewriting it in the simplest radical form. This means factoring the number under the square root into its prime factors, separating out any squares (since they can come out of the square root as their base), and rewriting the square root in a way that it's as simplified as it can be, combining any like terms if applicable. The variable 'p' will remain under the radical unless it's a perfect square or some factor of it is a perfect square.
$\sqrt{48 p}$
Answer
Solution:
Simplification Process:
Step 1:
Express $48p$ as a product of squares and other factors.
Step 1.1:
Identify that $48$ can be divided by $16$, which is a perfect square. Thus, we have $\sqrt{16 \cdot 3p}$.
Step 1.2:
Recognize that $16$ is the square of $4$, so write it as $\sqrt{4^2 \cdot 3p}$.
Step 1.3:
Further express $4$ as the square of $2$, leading to $\sqrt{(2^2)^2 \cdot 3p}$.
Step 1.4:
Enclose the expression in parentheses to emphasize the components under the radical: $\sqrt{((2^2)^2 \cdot (3p))}$.
Step 2:
Extract the square terms from under the square root, which gives us $2^2 \sqrt{3p}$.
Step 3:
Compute the square of $2$, which is $4$, to obtain the final simplified form: $4 \sqrt{3p}$.
Knowledge Notes:
To simplify a square root involving variables and constants, we can use the following steps:
Prime Factorization: Break down the number under the square root into its prime factors. This can help identify perfect squares that can be taken out of the square root.
Perfect Squares: Identify and separate perfect squares from the factors. A perfect square is a number that can be expressed as the square of an integer.
Simplify Radical: After separating the perfect squares, take their square root and place them outside the radical. The remaining factors that are not perfect squares stay under the radical.
Algebraic Rules: Apply algebraic rules for exponents, such as $(a^m)^n = a^{m \cdot n}$ and $\sqrt{a^2} = a$, to simplify the expression further.
Combining Like Terms: If there are like terms inside and outside the radical, combine them according to the rules of algebra.
In the given problem, we use these steps to simplify $\sqrt{48p}$. We factor 48 into $16 \cdot 3$ where 16 is a perfect square. We then express 16 as $4^2$ and further as $(2^2)^2$. After pulling out the perfect square, we are left with $4 \sqrt{3p}$ as the simplified form.
