What is the simplified form of [tex]$\sqrt{10,000 x^{64}}$[/tex] ?
A. [tex]$5000 x^{32}$[/tex]
B. [tex]$5000 x^8$[/tex]
C. [tex]$100 x^8$[/tex]
D. [tex]$100 x^{32}$[/tex]
Answer (1)
Find the square root of the constant term: 10000 = 100 .
Find the square root of the variable term: x 64 = x 32 .
Multiply the results: 100 × x 32 = 100 x 32 .
The simplified form is: 100 x 32 .
Explanation
Understanding the Problem We are asked to simplify the expression 10 , 000 x 64 . This involves finding the square root of both the constant term, 10,000, and the variable term, x 64 .
Square Root of the Constant First, let's find the square root of 10,000. The square root of 10,000 is 100, since 100 × 100 = 10 , 000 . So, 10 , 000 = 100 .
Square Root of the Variable Term Next, let's find the square root of x 64 . When taking the square root of a variable raised to a power, we divide the exponent by 2. Therefore, x 64 = x 2 64 = x 32 .
Combining the Results Now, we multiply the results from the previous two steps: 100 × x 32 = 100 x 32 .
Final Answer Therefore, the simplified form of 10 , 000 x 64 is 100 x 32 .
Examples
Imagine you are designing a square garden with an area of 10 , 000 x 64 square feet. To determine the length of one side of the garden, you need to find the square root of the area. This simplification helps you find that each side of the garden is 100 x 32 feet long. Understanding how to simplify such expressions is useful in various real-world scenarios involving areas and dimensions.
