Let's solve the given equation: [tex]\(\sqrt[m]{1000000} = 10\)[/tex]. Our objective is to find the value of [tex]\(m\)[/tex].
1. Translate the Root to Exponential Form:
The equation [tex]\(\sqrt[m]{1000000} = 10\)[/tex] can be rewritten using exponential notation:
[tex]\[
1000000^{1/m} = 10
\][/tex]
2. Raise Both Sides to the Power of [tex]\(m\)[/tex]:
If we raise both sides of the equation [tex]\(1000000^{1/m} = 10\)[/tex] to the power [tex]\(m\)[/tex], we get:
[tex]\[
(1000000^{1/m})^m = 10^m
\][/tex]
Simplifying the left side, the exponents [tex]\(1/m\)[/tex] and [tex]\(m\)[/tex] cancel out:
[tex]\[
1000000 = 10^m
\][/tex]
3. Express [tex]\(1000000\)[/tex] as a Power of [tex]\(10\)[/tex]:
We know that [tex]\(1000000\)[/tex] can be expressed as a power of [tex]\(10\)[/tex]:
[tex]\[
1000000 = 10^6
\][/tex]
4. Set the Expressions Equal:
Now that we have [tex]\(1000000 = 10^6\)[/tex], we can set the exponents equal to each other:
[tex]\[
10^m = 10^6
\][/tex]
5. Equate the Exponents:
Since the bases are the same, we can equate the exponents:
[tex]\[
m = 6
\][/tex]
6. Verification:
To verify, substitute [tex]\(m = 6\)[/tex] back into the original equation:
[tex]\[
\sqrt[6]{1000000} = 10
\][/tex]
As we calculated earlier, [tex]\(1000000 = 10^6\)[/tex], so:
[tex]\[
(10^6)^{1/6} = 10^{6 \cdot \frac{1}{6}} = 10^1 = 10
\][/tex]
This confirms that our solution is correct.
Thus, the value of [tex]\(m\)[/tex] is:
[tex]\[
\boxed{6}
\][/tex]
