Which expression is equivalent to $\sqrt{120 x} $?
A. $2 \sqrt{30 x}$
B. $2 \sqrt{30}$
C. $4 \sqrt{30 x}$
D. $60 \sqrt{2 x}$
Answer (2)
Factor 120 to find perfect square factors: 120 = 4 × 30 .
Rewrite the expression: 120 x = 4 × 30 x .
Use the property of square roots to simplify: 4 × 30 x = 4 × 30 x = 2 30 x .
The equivalent expression is 2 30 x .
Explanation
Understanding the problem We are given the expression 120 x and asked to find an equivalent expression from the given choices.
The choices are: 2 30 x 2 30 4 30 x 60 2 x
Plan To simplify the given expression, we need to factor the number inside the square root, 120, to find any perfect square factors.
Factoring 120 We can factor 120 as follows: 120 = 2 × 60 = 2 × 2 × 30 = 2 2 × 30 = 4 × 30 So, we can rewrite the original expression as: 120 x = 4 × 30 x
Simplifying the square root Using the property of square roots, ab = a × b , we can separate the factors: 4 × 30 x = 4 × 30 x = 2 30 x
Finding the equivalent expression Now we compare our simplified expression, 2 30 x , with the given choices. We see that it matches the first choice.
Therefore, the equivalent expression is 2 30 x .
Examples
Square roots appear in various contexts, such as calculating distances using the Pythagorean theorem. For example, if you have a right triangle with legs of length 30 x , then the length of the hypotenuse would be ( 30 x ) 2 + ( 30 x ) 2 = 2 ( 30 x ) = 60 x . Simplifying square roots helps in finding exact values and making calculations easier in geometry, physics, and engineering.
The expression 120 x simplifies to 2 30 x , which is equivalent to option A. This is done by factoring 120 into a perfect square and using square root properties. Therefore, the correct answer is A: 2 30 x .
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