What Is The Solution Of $(4x - 16)^{\frac{1}{2}} = 36$?A. X = 5 X = 5 X = 5 B. X = 13 X = 13 X = 13 C. X = 20 X = 20 X = 20 D. X = 328 X = 328 X = 328

Introduction

Solving equations involving exponents and radicals can be a challenging task in mathematics. In this article, we will focus on solving the equation $(4x - 16)^{\frac{1}{2}} = 36$. This equation involves a square root and a linear expression, making it a bit more complex than a standard linear equation. We will use algebraic techniques to simplify the equation and solve for the variable $x$.

Understanding the Equation

The given equation is $(4x - 16)^{\frac{1}{2}} = 36$. To start solving this equation, we need to understand the properties of exponents and radicals. The expression $(4x - 16)^{\frac{1}{2}}$ represents the square root of $4x - 16$. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we are looking for a value that, when squared, gives $4x - 16$.

Simplifying the Equation

To simplify the equation, we can start by isolating the radical expression. We can do this by squaring both sides of the equation. Squaring both sides will eliminate the radical sign and allow us to work with a simpler equation.

(4x - 16)^{\frac{1}{2}} = 36

Squaring both sides gives:

(4x - 16) = 1296

Solving for $x$

Now that we have simplified the equation, we can solve for $x$. We can start by adding $16$ to both sides of the equation to isolate the term with $x$.

4x - 16 + 16 = 1296 + 16

This simplifies to:

4x = 1312

Final Step

To solve for $x$, we can divide both sides of the equation by $4$.

\frac{4x}{4} = \frac{1312}{4}

This simplifies to:

x = 328

Conclusion

In this article, we solved the equation $(4x - 16)^{\frac{1}{2}} = 36$ using algebraic techniques. We started by understanding the properties of exponents and radicals, then simplified the equation by squaring both sides. Finally, we solved for $x$ by adding $16$ to both sides and dividing both sides by $4$. The solution to the equation is $x = 328$.

Answer

The correct answer is:

• D. $x = 328$

Discussion

This equation involves a square root and a linear expression, making it a bit more complex than a standard linear equation. The solution requires careful manipulation of the equation to isolate the variable $x$. The key steps in solving this equation are squaring both sides to eliminate the radical sign and adding $16$ to both sides to isolate the term with $x$.

Introduction

In our previous article, we solved the equation $(4x - 16)^{\frac{1}{2}} = 36$ using algebraic techniques. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the first step in solving the equation $(4x - 16)^{\frac{1}{2}} = 36$?

A: The first step in solving the equation is to understand the properties of exponents and radicals. We need to recognize that the expression $(4x - 16)^{\frac{1}{2}}$ represents the square root of $4x - 16$.

Q: Why do we need to square both sides of the equation?

A: We need to square both sides of the equation to eliminate the radical sign. Squaring both sides will allow us to work with a simpler equation and make it easier to solve for $x$.

Q: What is the next step after squaring both sides of the equation?

A: After squaring both sides of the equation, we need to add $16$ to both sides to isolate the term with $x$. This will help us to solve for $x$.

Q: How do we solve for $x$?

A: To solve for $x$, we need to divide both sides of the equation by $4$. This will give us the value of $x$.

Q: What is the final answer to the equation $(4x - 16)^{\frac{1}{2}} = 36$?

A: The final answer to the equation is $x = 328$.

Q: Can you provide an example of how to check the solution?

A: To check the solution, we can plug the value of $x$ back into the original equation and see if it is true. For example, we can substitute $x = 328$ into the equation $(4x - 16)^{\frac{1}{2}} = 36$ and see if it is equal to $36$.

Q: What are some common mistakes to avoid when solving equations with radicals?

A: Some common mistakes to avoid when solving equations with radicals include:

• Not squaring both sides of the equation
• Not adding or subtracting the same value to both sides of the equation
• Not checking the solution by plugging it back into the original equation

Q: Can you provide some tips for solving equations with radicals?

A: Some tips for solving equations with radicals include:

• Start by understanding the properties of exponents and radicals
• Square both sides of the equation to eliminate the radical sign
• Add or subtract the same value to both sides of the equation to isolate the term with the variable
• Check the solution by plugging it back into the original equation

Conclusion

In this Q&A article, we provided answers to common questions about solving the equation $(4x - 16)^{\frac{1}{2}} = 36$. We covered topics such as the first step in solving the equation, squaring both sides of the equation, and solving for $x$. We also provided tips and common mistakes to avoid when solving equations with radicals.