Questions: Simplify the following: sqrt(48) = To enter a sqrt(x) into your answer type sqrt(x)

Transcript text: Simplify the following: $\sqrt{48}=$ $\square$ To enter a $\sqrt{x}$ into you answer type sqrt( $x$ )

Solution

Solution Steps

To simplify $\sqrt{48}$, we need to find the prime factorization of 48 and then simplify the square root by pairing the prime factors. The goal is to express $\sqrt{48}$ in the form $a\sqrt{b}$ where $a$ and $b$ are integers and $b$ is as small as possible.

Step 1: Prime Factorization

To simplify $\sqrt{48}$, we first find its prime factorization. The number 48 can be expressed as: \[ 48 = 2^4 \times 3^1 \]

Step 2: Simplifying the Square Root

Next, we apply the property of square roots that states $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. We can separate the factors: \[ \sqrt{48} = \sqrt{2^4 \times 3^1} = \sqrt{(2^2)^2 \times 3} = \sqrt{(4^2) \times 3} \] This allows us to simplify: \[ \sqrt{48} = 2^2 \times \sqrt{3} = 4\sqrt{3} \]