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Which expression is equivalent to [tex]\sqrt{\sqrt{\sqrt{x^{48}}}}[/tex]?

A. [tex]x^6[/tex]
B. [tex]x^3[/tex]
C. [tex]x^{12}[/tex]
D. [tex]x^{24}[/tex]
E. [tex]x^{16}[/tex]


Sagot :

To find the expression that is equivalent to [tex]\(\sqrt{\sqrt{\sqrt{x^{48}}}}\)[/tex], we need to simplify the given expression step by step. Let's break it down:

1. Start with the innermost radical:
[tex]\[ \sqrt{x^{48}} \][/tex]
Rewriting the square root as an exponent, we have:
[tex]\[ (x^{48})^{1/2} \][/tex]
This simplifies to:
[tex]\[ x^{48 \cdot \frac{1}{2}} = x^{24} \][/tex]

2. Next, we take the square root of [tex]\(x^{24}\)[/tex]:
[tex]\[ \sqrt{x^{24}} \][/tex]
Rewriting the square root as an exponent, we get:
[tex]\[ (x^{24})^{1/2} \][/tex]
This simplifies to:
[tex]\[ x^{24 \cdot \frac{1}{2}} = x^{12} \][/tex]

3. Finally, we take the square root of [tex]\(x^{12}\)[/tex]:
[tex]\[ \sqrt{x^{12}} \][/tex]
Rewriting the square root as an exponent, we get:
[tex]\[ (x^{12})^{1/2} \][/tex]
This simplifies to:
[tex]\[ x^{12 \cdot \frac{1}{2}} = x^{6} \][/tex]

Therefore, the expression [tex]\(\sqrt{\sqrt{\sqrt{x^{48}}}}\)[/tex] simplifies to [tex]\(x^6\)[/tex].

So, the correct answer is:
[tex]\[ (A) \, x^6 \][/tex]