If x^(1/6) = 6, then (x^6)^(1/2) =

\(\sqrt[6]{x}=6\), raise both sides to the power \(6\)

\(=>x=6^6\). again raise both sides to the power \(6\)

\(=>x^6=(6^6)^6=6^{36}=(6^{18})^2\). Now take square root of both the sides

=>\(\sqrt{x^6}=\sqrt{(6^{18})^2}=6^{18}\)

Option D

\(\sqrt[6]{x}=6........x=6^6\)
\(\sqrt{x^6} = x^{\frac{6}{2}}=x^3=(6^6)^3=6^{18}\)
D
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\(\sqrt[6]{x}=6\)

So, \(x = 6^6\)

And, \(x^6 = 6^{36}\)

Thus, \(\sqrt{x^6} = 6^{18}\), answer will be (D)
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\(x^{(1/6)} = 6\)

\(x = 6^6\)

\(\sqrt{6^{36}}\)

\(6^{(18)}\)
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Raising both sides of the first equation to the 6th power, we have:

x = 6^6

So [(6^6)^6]^1/2 = (6^36)^1/2 = 6^18.

Answer: D
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hi there generis, today is International Exponents Day so happy day may you always have "positive exponents"

can you please explain one thing

for example when we raise both parts to the power of 6

\(=>x=6^6\) we multiply both sides by power 6

so we get \(x^6=6^{36}\)

now when i need to take a square root, does it mean that i need to multiply both sides by the exponent of \({1/2}\) , because radical sign means raised to the power of \(\frac{1}{2}\)
for example \(\sqrt{16}\) is the same as \(16^{\frac{1}{2}}\)

so taking square root from both parts \(x^6=6^{36}\) means \(x^6^{\frac{1}{2}}=6^{36}^{\frac{1}{2}}\) which is \(x^3=6^{18}\) ??

many thanks and happy sunday

Hi dave13 , yes, you are correct. "Take the square root" = raise the base to the power of \(\frac{1}{2}\). Often, the exponents get multiplied, just as you have done.

I think you wonder why your answer looks different. Its substance is NOT different.

These different forms of the expression all mean the same thing:
\(\sqrt{x^6}\)

\((x^6)^{\frac{1}{2}}\)

\(x^{(6*\frac{1}{2})}\)

\(x^3\)

Last one is the same because: \(x^{(6*\frac{1}{2})}=(x)^{((6*\frac{1}{2})=3)}=x^3\)

The prompt asked about this form of the term: \(\sqrt{x^6}\). So posters above stayed with that form. That's the only difference. Form.

Form can be tricky. You simplified both sides. The answer requires us to simplify only RHS.

Not having to simplify \(\sqrt{x^6}\) to \(x^3\) removed one extra step -- the one you took. But your math is correct.

Just watch the form in the question. We can simplify one side without simplifying the other. Simplifying does not change the value of the term.

(Multiplying one side by 2, e.g., does change the value. In that case we would have to multiply the other side by 2.)

I think I got to the heart of your question. If not, ask it a little more specifically. Hope that helps.

P.S. I like your exponent pun, hokey aspect and all. Made me grin. Your avatar makes perfect sense.
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