If , \(\sqrt{ 3^{n} } = 2187\) then the value of \( n \) is:
14
We are given the equation \( \sqrt{3^n} = 2187 \).
To solve for \(n\), we first square both sides of the equation to eliminate the square root:
\( (\sqrt{3^n})^2 = 2187^2 \)
\( 3^n = 2187^2 \)
Now, we need to find the prime factorization of 2187. We can do this by repeatedly dividing by 3:
2187 ÷ 3 = 729
729 ÷ 3 = 243
243 ÷ 3 = 81
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Therefore, the prime factorization of 2187 is \(3^7\).
Substituting this into our equation:
\( 3^n = (3^7)^2 \)
Using the property of exponents, \((a^b)^c = a^{bc}\), we get:
\( 3^n = 3^{14} \)
Since the bases are the same, we can equate the exponents:
\( n = 14 \)
Therefore, the value of \(n\) is 14.
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