Question

unit test
unit test complete
1 2
what is the simplified form of \\(\sqrt{10,000x^{64}}\\) ?
\\(\circ\\) \\(5000x^{32}\\)
\\(\circ\\) \\(5000x^{8}\\)
\\(\circ\\) \\(100x^{8}\\)
\\(\circ\\) \\(100x^{32}\\)
mark this and return
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Explanation:

Step1: Simplify the square root of 10,000

We know that \( \sqrt{10000} = 100 \) because \( 100\times100 = 10000 \).

Step2: Simplify the square root of \( x^{64} \)

Using the property of exponents \( \sqrt{a^m}=a^{\frac{m}{2}} \), for \( a = x \) and \( m = 64 \), we have \( \sqrt{x^{64}}=x^{\frac{64}{2}}=x^{32} \)? Wait, no, wait. Wait, maybe I made a mistake. Wait, no, the options have \( x^{8} \) or \( x^{32} \) or \( x^{22} \). Wait, no, let's re - check. Wait, the original expression is \( \sqrt{10000x^{64}}=\sqrt{10000}\times\sqrt{x^{64}} \). We know that \( \sqrt{10000} = 100 \), and \( \sqrt{x^{64}}=x^{\frac{64}{2}}=x^{32} \)? But the options have \( 100x^{32} \) as one of the options? Wait, the options are:

1. \( 5000x^{32} \)
2. \( 5000x^{8} \)
3. \( 100x^{8} \)
4. \( 100x^{32} \)

Wait, maybe I misread the exponent. Wait, the problem is \( \sqrt{10000x^{64}} \)? Or is it \( \sqrt{10000x^{16}} \)? Wait, no, the user's problem says \( x^{64} \). Wait, but if we look at the options, \( 100x^{32} \) would be correct if the exponent is 64. But let's check again.

Wait, \( \sqrt{10000}=100 \), and \( \sqrt{x^{64}} = x^{32} \) (since \( (x^{32})^2=x^{64} \)). So \( \sqrt{10000x^{64}}=100x^{32} \), which is option D. But wait, maybe the exponent is 16? Wait, no, the user's problem says \( x^{64} \). Wait, let's check the options again. The options are:

• \( 5000x^{32} \)
• \( 5000x^{8} \)
• \( 100x^{8} \)
• \( 100x^{32} \)

So the correct step - by - step:

Step1: Simplify \( \sqrt{10000} \)

\( \sqrt{10000}=100 \) because \( 100\times100 = 10000 \).

Step2: Simplify \( \sqrt{x^{64}} \)

Using the formula \( \sqrt{a^m}=a^{\frac{m}{2}} \) (for \( a\geq0 \)), when \( a = x \) and \( m = 64 \), we have \( \sqrt{x^{64}}=x^{\frac{64}{2}}=x^{32} \).

Step3: Multiply the two results

\( \sqrt{10000x^{64}}=\sqrt{10000}\times\sqrt{x^{64}} = 100\times x^{32}=100x^{32} \)

Answer:

\( 100x^{32} \) (corresponding to the option with \( 100x^{32} \))