Select the correct answer.
Which expression is equivalent to the given expression?
[tex]$\sqrt{45}$[/tex]
A. [tex]$9 \sqrt{5}$[/tex]
B. [tex]$5 \sqrt{9}$[/tex]
C. [tex]$5 \sqrt{3}$[/tex]
D. [tex]$3 \sqrt{5}$[/tex]
Answer (2)
Find the prime factorization of 45, which is 3 2 × 5 .
Rewrite the square root as 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 given options. To do this, we need to simplify the square root.
Prime factorization We can simplify 45 by finding the prime factorization of 45. The prime factorization of 45 is 3 × 3 × 5 , which can be written as 3 2 × 5 .
Rewriting the square root Now we can rewrite the square root as 45 = 3 2 × 5 . Using the property of square roots, we can separate the factors: 3 2 × 5 = 3 2 × 5 .
Simplifying the square root Since 3 2 = 3 , we have 45 = 3 5 .
Finding the correct option Comparing our simplified expression 3 5 with the given options, we see that option D matches our result. Therefore, the correct answer is D.
Examples
Square roots are used in many real-world applications, 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 you have a square garden with an area of 45 square meters, you can use the square root of 45 to find the length of one side of the garden, which is approximately 6.71 meters. Simplifying square roots helps in obtaining more manageable and understandable values for these calculations.
To simplify 45 , we find that it equals 3 5 . The correct answer from the options is therefore D. 3 5 .
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