What is the simplified form of [tex]$\sqrt{400 x^{100}}$[/tex] ?
A. [tex]$200 x^{10}$[/tex]
B. [tex]$200 x^{50}$[/tex]
C. [tex]$20 x^{10}$[/tex]
D. [tex]$20 x^{50}$[/tex]
Answer (2)
Separate the square root: 400 x 100 = 400 ⋅ x 100 .
Calculate the square root of 400: 400 = 20 .
Calculate the square root of x 100 : x 100 = x 50 .
Combine the results: The simplified form is 20 x 50 .
Explanation
Understanding the problem We are asked to simplify the expression 400 x 100 . This involves finding the square root of both the constant term and the variable term.
Separating the terms We can rewrite the expression as a product of square roots: 400 x 100 = 400 ⋅ x 100
Square root of the constant First, let's find the square root of 400. Since 20 × 20 = 400 , we have 400 = 20
Square root of the variable term Next, let's find the square root of x 100 . Recall that x n = x n /2 . Therefore, x 100 = x 100/2 = x 50
Combining the results Now, we combine the results: 400 x 100 = 400 ⋅ x 100 = 20 ⋅ x 50 = 20 x 50 So, the simplified form of 400 x 100 is 20 x 50 .
Examples
Imagine you are calculating the area of a square, and you find that the area is given by 400 x 100 . To find the side length of the square, you need to take the square root of the area. Simplifying 400 x 100 to 20 x 50 gives you the side length of the square. This type of simplification is also useful in physics when dealing with quantities that involve square roots, such as calculating velocities or energies.
The simplified form of 400 x 100 is 20 x 50 , which is calculated by simplifying the square root of both the constant and the variable term. Therefore, the correct option is D: 20 x 50 .
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