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Solve for [tex]x[/tex]:

[tex]\[
\left(\frac{1}{x}\right)^a = x^3
\][/tex]


Sagot :

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.