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22 Feb 2023, 06:03
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If \(n\) is an integer greater than 1, what is the value of \(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\), where the given expression extends to an infinite number of roots?
A. \(10\)
B. \(10^{\frac{1}{n}}\)
C. \(10^{\frac{n-1}{n}}\)
D. \(10^{\frac{n}{n-1}}\)
E. \(10^{n}\)
A. \(10\)
B. \(10^{\frac{1}{n}}\)
C. \(10^{\frac{n-1}{n}}\)
D. \(10^{\frac{n}{n-1}}\)
E. \(10^{n}\)
22 Feb 2023, 06:03
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Official Solution:
If \(n\) is an integer greater than 1, what is the value of \(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\), where the given expression extends to an infinite number of roots?
A. \(10\)
B. \(10^{\frac{1}{n}}\)
C. \(10^{\frac{n-1}{n}}\)
D. \(10^{\frac{n}{n-1}}\)
E. \(10^{n}\)
Let \(x=10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\)
Now, re-write above as \(x=10*\sqrt[n]{(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...})}}}\).
Since the expression extends to an infinite number of roots, then the expression in brackets would also equal to \(x\). Thus we can replace the expression in brackets with \(x\) and rewrite the expression as: \(x=10*\sqrt[n]{x}\)
Take above to the \(n^{th}\) power:
\(x^n=10^n*x\)
\(x^{n-1}=10^n\)
Take \(n-1^{th}\) root:
\(x=10^{\frac{n}{n-1}}\)
Answer: D
If \(n\) is an integer greater than 1, what is the value of \(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\), where the given expression extends to an infinite number of roots?
A. \(10\)
B. \(10^{\frac{1}{n}}\)
C. \(10^{\frac{n-1}{n}}\)
D. \(10^{\frac{n}{n-1}}\)
E. \(10^{n}\)
Let \(x=10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...}}}}\)
Now, re-write above as \(x=10*\sqrt[n]{(10*\sqrt[n]{10*\sqrt[n]{10*\sqrt[n]{...})}}}\).
Since the expression extends to an infinite number of roots, then the expression in brackets would also equal to \(x\). Thus we can replace the expression in brackets with \(x\) and rewrite the expression as: \(x=10*\sqrt[n]{x}\)
Take above to the \(n^{th}\) power:
\(x^n=10^n*x\)
\(x^{n-1}=10^n\)
Take \(n-1^{th}\) root:
\(x=10^{\frac{n}{n-1}}\)
Answer: D
gmat_prep_med
12 Feb 2025, 23:22
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Hi,
Why does the expression in bracket equal to x when there are infinite roots?
Why does the expression in bracket equal to x when there are infinite roots?
Bunuel
