Solve The Equation:$(3x - 4)^{\frac{2}{5}} = 4$

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Introduction

In this article, we will delve into solving the equation $(3x - 4)^{\frac{2}{5}} = 4$. This equation involves a fractional exponent, which can be challenging to solve. We will break down the solution step by step, using algebraic manipulations and properties of exponents.

Step 1: Isolate the Variable

The first step in solving the equation is to isolate the variable $x$. To do this, we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the power of $\frac{5}{2}$, which is the reciprocal of the exponent.

(3x - 4)^{\frac{2}{5}} = 4

Raising both sides to the power of $\frac{5}{2}$:

(3x - 4)^{\frac{2}{5}} \cdot (3x - 4)^{\frac{5}{2}} = 4 \cdot (3x - 4)^{\frac{5}{2}}

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the left-hand side:

(3x - 4)^{\frac{2}{5} + \frac{5}{2}} = 4 \cdot (3x - 4)^{\frac{5}{2}}

Simplifying the exponent:

(3x - 4)^{\frac{9}{10}} = 4 \cdot (3x - 4)^{\frac{5}{2}}

Step 2: Simplify the Equation

Now that we have isolated the variable, we can simplify the equation by getting rid of the fraction on the right-hand side. We can do this by multiplying both sides of the equation by $(3x - 4)^{-\frac{5}{2}}$.

(3x - 4)^{\frac{9}{10}} = 4 \cdot (3x - 4)^{\frac{5}{2}}

Multiplying both sides by $(3x - 4)^{-\frac{5}{2}}$:

(3x - 4)^{\frac{9}{10}} \cdot (3x - 4)^{-\frac{5}{2}} = 4 \cdot (3x - 4)^{\frac{5}{2}} \cdot (3x - 4)^{-\frac{5}{2}}

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the left-hand side:

(3x - 4)^{\frac{9}{10} - \frac{5}{2}} = 4 \cdot (3x - 4)^{\frac{5}{2} - \frac{5}{2}}

Simplifying the exponent:

(3x - 4)^{\frac{9}{10} - \frac{5}{2}} = 4 \cdot (3x - 4)^0

Simplifying the right-hand side:

(3x - 4)^{\frac{9}{10} - \frac{5}{2}} = 4

Step 3: Solve for x

Now that we have simplified the equation, we can solve for $x$. We can do this by isolating the variable $x$.

(3x - 4)^{\frac{9}{10} - \frac{5}{2}} = 4

Using the property of exponents that states $a^m = a^n$ if and only if $m = n$, we can equate the exponents:

\frac{9}{10} - \frac{5}{2} = 0

Simplifying the equation:

\frac{9}{10} = \frac{5}{2}

Cross-multiplying:

18 = 25

This is a contradiction, which means that the original equation has no solution.

Conclusion

In this article, we solved the equation $(3x - 4)^{\frac{2}{5}} = 4$. We broke down the solution into three steps: isolating the variable, simplifying the equation, and solving for $x$. We found that the original equation has no solution.

Final Answer

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Introduction

In our previous article, we solved the equation $(3x - 4)^{\frac{2}{5}} = 4$. We broke down the solution into three steps: isolating the variable, simplifying the equation, and solving for $x$. We found that the original equation has no solution. In this article, we will answer some frequently asked questions about the solution.

Q: What is the fractional exponent in the equation?

A: The fractional exponent in the equation is $\frac{2}{5}$. This means that the base $(3x - 4)$ is raised to the power of $\frac{2}{5}$.

Q: Why did we raise both sides of the equation to the power of $\frac{5}{2}$?

A: We raised both sides of the equation to the power of $\frac{5}{2}$ to get rid of the fractional exponent. This is because the reciprocal of the exponent $\frac{2}{5}$ is $\frac{5}{2}$.

Q: What is the property of exponents that we used to simplify the left-hand side of the equation?

A: The property of exponents that we used to simplify the left-hand side of the equation is $a^m \cdot a^n = a^{m+n}$.

Q: Why did we multiply both sides of the equation by $(3x - 4)^{-\frac{5}{2}}$?

A: We multiplied both sides of the equation by $(3x - 4)^{-\frac{5}{2}}$ to get rid of the fraction on the right-hand side.

Q: What is the property of exponents that we used to simplify the right-hand side of the equation?

A: The property of exponents that we used to simplify the right-hand side of the equation is $a^m = a^n$ if and only if $m = n$.

Q: Why did we find that the original equation has no solution?

A: We found that the original equation has no solution because the equation $(3x - 4)^{\frac{9}{10} - \frac{5}{2}} = 4$ is a contradiction.

Q: What is the final answer to the equation?

A: The final answer to the equation is $\boxed{No solution}$.

Common Mistakes

• Not isolating the variable before simplifying the equation.
• Not using the correct property of exponents to simplify the equation.
• Not checking for contradictions before solving for $x$.

Tips and Tricks

• Always isolate the variable before simplifying the equation.
• Use the correct property of exponents to simplify the equation.
• Check for contradictions before solving for $x$.

Conclusion

In this article, we answered some frequently asked questions about the solution to the equation $(3x - 4)^{\frac{2}{5}} = 4$. We covered topics such as fractional exponents, properties of exponents, and common mistakes. We hope that this article has been helpful in understanding the solution to the equation.