Simplify the following radical expression.
$\sqrt{48}$
A. $16 \sqrt{3}$
B. $4 \sqrt{3}$
C. $4 \sqrt{2}$
D. $6 \sqrt{2}$
Answer (2)
∙ Find the prime factorization of 48: 48 = 2 4 × 3 .$\bullet R e w r i t e t h es q u a reroo t : \sqrt{48} = \sqrt{2^4 \times 3} .$ ∙ Separate the factors: 2 4 × 3 = 2 4 × 3 .$\bullet S im pl i f y : \sqrt{48} = 4 \sqrt{3} . \The s im pl i f i e d r a d i c a l e x p ress i o ni s \boxed{4 \sqrt{3}}$.
Explanation
Understanding the problem We are asked to simplify the radical expression 48 . This means we want to find the largest perfect square that divides 48 and take its square root out of the radical.
Prime factorization of 48 First, we find the prime factorization of 48. We can write 48 as 48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3 = 2 4 × 3 .
Rewriting the radical Now we can rewrite the square root of 48 as 48 = 2 4 × 3 .
Separating the factors Using the property of square roots, we can separate the factors: 2 4 × 3 = 2 4 × 3 .
Simplifying the radical Since 2 4 = 16 , we have 2 4 = 16 = 4 . Therefore, 48 = 4 3 .
Examples
Radical expressions are used in many areas of math and science. For example, when calculating the distance between two points in a coordinate plane, we often end up with a radical expression. Simplifying these expressions allows us to work with more manageable numbers and gain a better understanding of the relationships between different quantities. Imagine you're building a square garden and need to determine the length of the diagonal. If the side of the garden is 8 meters, the diagonal would be 8 × 2 = 16 = 4 meters. Simplifying radicals helps in practical measurements and designs.
The simplified form of 48 is 4 3 , which we obtain by finding the prime factorization, separating the factors under the square root, and simplifying. Therefore, the answer is Option B . This technique of simplifying radicals helps make expressions easier to understand and work with.
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