Alright, let's carefully analyze and solve the given mathematical equation step-by-step.
We start with the equation:
[tex]\[
\sqrt{b^m} = \left( \sqrt[n]{b} \right)^m
\][/tex]
### Step 1: Simplify the Left-Hand Side (LHS)
The left-hand side of the equation is:
[tex]\[
\sqrt{b^m}
\][/tex]
The square root of [tex]\( b^m \)[/tex] can be rewritten using the exponent rule that [tex]\( \sqrt{x} = x^{1/2} \)[/tex]. Thus:
[tex]\[
\sqrt{b^m} = (b^m)^{1/2}
\][/tex]
Applying the power of a power property, [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(b^m)^{1/2} = b^{m \cdot \frac{1}{2}} = b^{m/2}
\][/tex]
So, the simplified left-hand side is:
[tex]\[
b^{m/2}
\][/tex]
### Step 2: Simplify the Right-Hand Side (RHS)
The right-hand side of the equation is:
[tex]\[
\left( \sqrt[n]{b} \right)^m
\][/tex]
The [tex]\(n\)[/tex]-th root of [tex]\(b\)[/tex] can be written using exponents as [tex]\( \sqrt[n]{b} = b^{1/n} \)[/tex]. Thus:
[tex]\[
\left( \sqrt[n]{b} \right)^m = (b^{1/n})^m
\][/tex]
Again applying the power of a power property:
[tex]\[
(b^{1/n})^m = b^{(1/n) \cdot m} = b^{m/n}
\][/tex]
So, the simplified right-hand side is:
[tex]\[
b^{m/n}
\][/tex]
### Step 3: Compare Both Sides
We now have the simplified forms of both sides of the original equation:
Left-hand side:
[tex]\[
b^{m/2}
\][/tex]
Right-hand side:
[tex]\[
b^{m/n}
\][/tex]
For the equation to hold true, the exponents must be equal:
[tex]\[
\frac{m}{2} = \frac{m}{n}
\][/tex]
### Step 4: Solve for [tex]\( n \)[/tex]
To solve the above equality, we eliminate the common factor [tex]\( m \)[/tex] (assuming [tex]\( m \neq 0 \)[/tex]):
[tex]\[
\frac{1}{2} = \frac{1}{n}
\][/tex]
Taking the reciprocal of both sides:
[tex]\[
2 = n
\][/tex]
### Conclusion
The equation [tex]\( \sqrt{b^m} = \left( \sqrt[n]{b} \right)^m \)[/tex] holds true if and only if [tex]\( n = 2 \)[/tex].
