Solve For { X $}$ In The Equation:${ \frac{1}{3} X^5 = 81 }$

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Introduction

Solving for ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$ requires a combination of algebraic manipulation and exponent rules. This equation involves a variable raised to the power of 5, making it a quintic equation. In this article, we will guide you through the step-by-step process of solving for ${ x }$ and provide a clear understanding of the mathematical concepts involved.

Understanding the Equation

The given equation is ${ \frac{1}{3} x^5 = 81 }$. To solve for ${ x }$, we need to isolate the variable ${ x }$ on one side of the equation. The first step is to get rid of the fraction by multiplying both sides of the equation by 3.

Multiplying Both Sides by 3

${ \frac{1}{3} x^5 = 81 }$

Multiplying both sides by 3 gives us:

${ x^5 = 243 }$

Isolating the Variable

Now that we have ${ x^5 = 243 }$, we need to isolate the variable ${ x }$. To do this, we take the fifth root of both sides of the equation.

Taking the Fifth Root

Taking the fifth root of both sides gives us:

${ x = \sqrt[5]{243} }$

Simplifying the Expression

The expression ${ \sqrt[5]{243} }$ can be simplified further. We know that ${ 243 = 3^5 }$, so we can rewrite the expression as:

${ x = \sqrt[5]{3^5} }$

Using the property of exponents that states ${ \sqrt[n]{a^n} = a }$, we can simplify the expression to:

${ x = 3 }$

Conclusion

In this article, we solved for ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$ by following a step-by-step process. We started by getting rid of the fraction, then isolated the variable by taking the fifth root of both sides. Finally, we simplified the expression to find the value of ${ x }$. The solution to the equation is ${ x = 3 }$.

Additional Tips and Tricks

When solving equations involving exponents, it's essential to remember the following tips and tricks:

• Exponent rules: When dealing with exponents, remember that ${ a^m \cdot a^n = a^{m+n} }$ and ${ \frac{a^m}{a^n} = a^{m-n} }$.
• Roots: When taking the nth root of a number, remember that ${ \sqrt[n]{a^n} = a }$.
• Simplifying expressions: When simplifying expressions involving exponents, look for opportunities to combine like terms and use exponent rules to simplify the expression.

Frequently Asked Questions

• What is the value of ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$?
• How do I solve equations involving exponents?
• What are some common exponent rules that I should remember?

Final Thoughts

Solving for ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$ requires a combination of algebraic manipulation and exponent rules. By following the step-by-step process outlined in this article, you can solve for ${ x }$ and gain a deeper understanding of the mathematical concepts involved. Remember to always simplify expressions and use exponent rules to make solving equations easier and more efficient.

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Introduction

In our previous article, we solved for ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$ by following a step-by-step process. However, we know that there are often many questions and concerns that arise when dealing with mathematical equations. In this article, we will address some of the most frequently asked questions related to solving for ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$.

Q&A

Q: What is the value of ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$?

A: The value of ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$ is ${ x = 3 }$.

Q: How do I solve equations involving exponents?

A: To solve equations involving exponents, follow these steps:

1. Get rid of any fractions by multiplying both sides of the equation by the denominator.
2. Isolate the variable by taking the nth root of both sides of the equation.
3. Simplify the expression by combining like terms and using exponent rules.

Q: What are some common exponent rules that I should remember?

A: Some common exponent rules that you should remember include:

• ${ a^m \cdot a^n = a^{m+n} }$
• ${ \frac{a^m}{a^n} = a^{m-n} }$
• ${ \sqrt[n]{a^n} = a }$

Q: How do I simplify expressions involving exponents?

A: To simplify expressions involving exponents, look for opportunities to combine like terms and use exponent rules to simplify the expression. For example, if you have the expression ${ 2^3 \cdot 2^2 }$, you can simplify it by combining the exponents: ${ 2^3 \cdot 2^2 = 2^{3+2} = 2^5 }$.

Q: What if I have a negative exponent in the equation?

A: If you have a negative exponent in the equation, you can rewrite it as a positive exponent by taking the reciprocal of both sides of the equation. For example, if you have the equation ${ 2^{-3} = 4 }$, you can rewrite it as ${ 2^3 = \frac{1}{4} }$.

Q: Can I use a calculator to solve equations involving exponents?

A: Yes, you can use a calculator to solve equations involving exponents. However, it's essential to remember that calculators can only give you an approximate value for the solution. To get an exact value, you need to follow the step-by-step process outlined in this article.

Conclusion

In this article, we addressed some of the most frequently asked questions related to solving for ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$. We provided step-by-step instructions on how to solve equations involving exponents and highlighted some common exponent rules that you should remember. By following these tips and tricks, you can become more confident and proficient in solving equations involving exponents.

Additional Resources

• Mathematical formulas and equations: For a comprehensive list of mathematical formulas and equations, visit our website or consult a mathematics textbook.
• Exponent rules: For a detailed explanation of exponent rules, visit our website or consult a mathematics textbook.
• Solving equations involving exponents: For a step-by-step guide on how to solve equations involving exponents, visit our website or consult a mathematics textbook.

Final Thoughts

Solving for ${ x }$ in the equation ${ \frac{1}{3} x^5 = 81 }$ requires a combination of algebraic manipulation and exponent rules. By following the step-by-step process outlined in this article and remembering some common exponent rules, you can become more confident and proficient in solving equations involving exponents.