Let's solve the equation [tex]\(3^{x^3} = 243\)[/tex] step-by-step.
1. Recognize the Form of the Equation:
The given equation is [tex]\(3^{x^3} = 243\)[/tex].
2. Express 243 as a Power of 3:
To simplify the comparison, express [tex]\(243\)[/tex] as a power of [tex]\(3\)[/tex]. We know that:
[tex]\[
243 = 3^5
\][/tex]
So, we can rewrite the equation as:
[tex]\[
3^{x^3} = 3^5
\][/tex]
3. Match the Exponents:
Since the bases are the same, we can equate the exponents:
[tex]\[
x^3 = 5
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], take the cube root of both sides of the equation:
[tex]\[
x = \sqrt[3]{5}
\][/tex]
5. Express the Solution Numerically:
The cube root of [tex]\(5\)[/tex] is approximately:
[tex]\[
x \approx 1.7099759466766968
\][/tex]
Thus, the solution to the equation [tex]\(3^{x^3} = 243\)[/tex] is:
[tex]\[
x \approx 1.7099759466766968
\][/tex]
