What is the square root of 16x^16
The correct answser is: 4x^8
The Correct Answer and Explanation is :
The square root of ( 16x^{16} ) can be simplified as follows:
[
\sqrt{16x^{16}} = \sqrt{16} \times \sqrt{x^{16}}
]
- Square root of 16:
[
\sqrt{16} = 4
]
This is because ( 4 \times 4 = 16 ). - Square root of ( x^{16} ):
[
\sqrt{x^{16}} = x^{16/2} = x^8
]
When taking the square root of a power, you divide the exponent by 2.
Therefore, the square root of ( 16x^{16} ) is:
[
4x^8
]
Explanation (in 300 words):
The square root function undoes squaring, which means that if you square a number and then take the square root of that result, you get back the original number (as long as we are dealing with non-negative values). In this problem, we are dealing with the expression ( 16x^{16} ), which consists of two parts: a constant ( 16 ) and a variable term ( x^{16} ).
Step 1: Simplifying the constant
The square root of ( 16 ) is straightforward. We know that ( \sqrt{16} = 4 ) because multiplying 4 by itself yields 16 (i.e., ( 4 \times 4 = 16 )).
Step 2: Simplifying the variable part
For the variable part, we have ( x^{16} ). The square root of ( x^{16} ) follows the rule for exponents under a square root:
[
\sqrt{x^n} = x^{n/2}
]
So, applying this rule to ( x^{16} ), we divide the exponent (16) by 2, which gives us ( x^8 ).
Step 3: Combining the results
Finally, we combine the two simplified components: ( \sqrt{16} = 4 ) and ( \sqrt{x^{16}} = x^8 ), giving the final simplified expression:
[
4x^8
]
This shows that the square root of ( 16x^{16} ) is ( 4x^8 ), following basic rules of algebra involving square roots and exponents.