Question
Solve for
\[n\].
\[\left(z^2\right)^{n}=z^{10}\]
\[n=\]
\[n\].
\[\left(z^2\right)^{n}=z^{10}\]
\[n=\]
Answers
Answered by
GPT-4o mini
To solve the equation
\[
(z^2)^n = z^{10},
\]
we can apply the power rule of exponents which states that \((a^m)^n = a^{mn}\). This allows us to simplify the left side of the equation:
\[
(z^2)^n = z^{2n}.
\]
Thus, we can rewrite the equation as:
\[
z^{2n} = z^{10}.
\]
Since the bases (z) are the same, we can now equate the exponents:
\[
2n = 10.
\]
To solve for \(n\), we can divide both sides of the equation by 2:
\[
n = \frac{10}{2} = 5.
\]
Thus, the solution for \(n\) is
\[
\boxed{5}.
\]
\[
(z^2)^n = z^{10},
\]
we can apply the power rule of exponents which states that \((a^m)^n = a^{mn}\). This allows us to simplify the left side of the equation:
\[
(z^2)^n = z^{2n}.
\]
Thus, we can rewrite the equation as:
\[
z^{2n} = z^{10}.
\]
Since the bases (z) are the same, we can now equate the exponents:
\[
2n = 10.
\]
To solve for \(n\), we can divide both sides of the equation by 2:
\[
n = \frac{10}{2} = 5.
\]
Thus, the solution for \(n\) is
\[
\boxed{5}.
\]
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