At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
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.
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.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.