If 100! 100! 100! 99! 99! 99!= [8]10^x, then what is the value of x?

If 100! 100! 100! 99! 99! 99!= [8]10^x, then… | GMAT Sprint

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PS · Problem Solving

705-805

Roots

If $\frac{\sqrt{100!\sqrt{100!\sqrt{100!}}}}{\sqrt{99!\sqrt{99!\sqrt{99!}}}}=\sqrt[8]{10^x}$ , then what is the value of x?

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Explanation

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Get rid of the roots:

\frac { (100!)^{\frac{1}{2}}*(100!)^{\frac{1}{4}}*(100!)^{\frac{1}{8}} } {(99!)^{\frac{1}{2}}*(99!)^{\frac{1}{4}}*(99!)^{\frac{1}{8}} } =\sqrt[8]{10^x}

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Take the whole expression to the eights power:

\left(\frac { (100!)^{\frac{1}{2}}*(100!)^{\frac{1}{4}}*(100!)^{\frac{1}{8}} } {(99!)^{\frac{1}{2}}*(99!)^{\frac{1}{4}}*(99!)^{\frac{1}{8}} } \right)^8=(\sqrt[8]{10^x})^8

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What this question tests

Your handling of square and higher roots, including simplifying radicals.

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.