Alright, let's solve the equation [tex]\(\left(\frac{1}{x}\right)^a = x^3\)[/tex] step-by-step.
1. Simplify the left-hand side:
[tex]\[\left(\frac{1}{x}\right)^a\][/tex]
This can be rewritten using the properties of exponents:
[tex]\[\left(\frac{1}{x}\right)^a = \frac{1^a}{x^a} = \frac{1}{x^a}\][/tex]
So the equation now becomes:
[tex]\[\frac{1}{x^a} = x^3\][/tex]
2. Clear the fraction by multiplying both sides by [tex]\(x^a\)[/tex]:
[tex]\[\frac{1}{x^a} \cdot x^a = x^3 \cdot x^a\][/tex]
This simplifies to:
[tex]\[1 = x^{3 + a}\][/tex]
3. Find the solution for [tex]\(x\)[/tex]:
For the equation [tex]\(1 = x^{3 + a}\)[/tex] to hold true, [tex]\(x^{3 + a}\)[/tex] must be equal to 1. There are specific values of [tex]\(x\)[/tex] that satisfy this equation, depending on the value of the exponent [tex]\(3 + a\)[/tex]:
If [tex]\(x = 1\)[/tex]:
[tex]\[1^{3 + a} = 1\][/tex]
This holds true for any value of [tex]\(a\)[/tex].
Consequently, [tex]\(x = 1\)[/tex] is a valid solution to the equation.
Thus, the solution to the equation [tex]\(\left(\frac{1}{x}\right)^a = x^3\)[/tex] is:
[tex]\[
x = 1
\][/tex]
Therefore, [tex]\(x = 1\)[/tex] is the value that satisfies the original equation.
