Select the correct answer.
Which expression is equivalent to the given expression?
[tex]$\sqrt{45}$[/tex]
A. [tex]$5 \sqrt{3}$[/tex]
B. [tex]$3 \sqrt{5}$[/tex]
C. [tex]$9 \sqrt{5}$[/tex]
D. [tex]$5 \sqrt{9}$[/tex]
Answer (2)
Find the prime factorization of 45: 45 = 3 2 × 5 .
Rewrite the square root: 45 = 3 2 × 5 .
Simplify the square root: 45 = 3 2 × 5 = 3 5 .
The equivalent expression is: 3 5 .
Explanation
Understanding the Problem We are given the expression 45 and asked to find an equivalent expression from the options: A. 5 3 , B. 3 5 , C. 9 5 , D. 5 9 . To do this, we need to simplify 45 .
Prime Factorization To simplify 45 , we first find the prime factorization of 45. We can write 45 = 9 × 5 = 3 × 3 × 5 = 3 2 × 5 .
Rewriting the Square Root Now we can rewrite the square root expression as 45 = 3 2 × 5 . Using the property of square roots, we can separate the factors: 3 2 × 5 = 3 2 × 5 .
Simplifying Since 3 2 = 3 , we have 45 = 3 × 5 = 3 5 .
Finding the Correct Option Comparing our simplified expression 3 5 with the given options, we see that it matches option B.
Examples
Square roots appear in various contexts, such as calculating the distance between two points in a coordinate plane or determining the length of the side of a square given its area. For example, if a square has an area of 45 square units, the length of each side is 45 units, which simplifies to 3 5 units. Understanding how to simplify square roots helps in practical applications like these.
The expression 45 simplifies to 3 5 . Therefore, the correct answer is option B. 3 5 .
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