Select The Correct Answer.What Is The Solution To The Equation? $\sqrt[4]{x-4}=3$A. -8 B. 16 C. 85 D. No Solution

Solving the Equation: Uncovering the Solution to $\sqrt[4]{x-4}=3$

In mathematics, equations involving roots and radicals can be challenging to solve. The equation $\sqrt[4]{x-4}=3$ is a great example of such a problem. In this article, we will delve into the world of radical equations and explore the solution to this particular equation. We will examine the properties of radicals, the steps involved in solving radical equations, and finally, we will arrive at the correct solution.

Before we dive into solving the equation, let's take a moment to understand the concept of radicals. A radical is a mathematical expression that involves a root or a power of a number. The most common radicals are square roots ($\sqrt{x}$), cube roots ($\sqrt[3]{x}$), and fourth roots ($\sqrt[4]{x}$). In this case, we are dealing with a fourth root.

Properties of Radicals

Radicals have several properties that are essential to understand when solving equations involving radicals. Some of these properties include:

• Multiplication Property: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$
• Division Property: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
• Power Property: $(\sqrt[n]{a})^m = \sqrt[nm]{a^m}$

Now that we have a good understanding of radicals, let's move on to solving the equation $\sqrt[4]{x-4}=3$. To solve this equation, we will use the following steps:

1. Raise both sides to the power of 4: This will eliminate the fourth root on the left-hand side of the equation.
2. Simplify the equation: We will simplify the equation by evaluating the expression on the left-hand side.
3. Solve for x: Finally, we will solve for x by isolating the variable on one side of the equation.

Step 1: Raise both sides to the power of 4

To eliminate the fourth root on the left-hand side of the equation, we will raise both sides to the power of 4. This gives us:

$$\left(\sqrt[4]{x-4}\right)^4 = 3^4$$

Using the power property of radicals, we can simplify the left-hand side of the equation:

$$x-4 = 81$$

Step 2: Simplify the equation

Now that we have simplified the equation, let's isolate the variable x by adding 4 to both sides of the equation:

$$x = 81 + 4$$

Step 3: Solve for x

Finally, we can solve for x by evaluating the expression on the right-hand side of the equation:

$$x = 85$$

In conclusion, the solution to the equation $\sqrt[4]{x-4}=3$ is x = 85. We arrived at this solution by using the properties of radicals, raising both sides of the equation to the power of 4, simplifying the equation, and finally, solving for x.

The correct answer is C. 85.
Frequently Asked Questions: Solving Radical Equations

In our previous article, we explored the solution to the equation $\sqrt[4]{x-4}=3$. We delved into the world of radical equations and uncovered the properties of radicals, the steps involved in solving radical equations, and finally, we arrived at the correct solution. In this article, we will address some of the most frequently asked questions related to solving radical equations.

Q: What is the difference between a square root and a fourth root?

A: A square root is a mathematical expression that involves a root or a power of a number, specifically the second root. It is denoted by the symbol $\sqrt{x}$. On the other hand, a fourth root is a mathematical expression that involves a root or a power of a number, specifically the fourth root. It is denoted by the symbol $\sqrt[4]{x}$.

Q: How do I know which root to use when solving an equation?

A: When solving an equation involving a root, you need to determine which root is being used. In the case of the equation $\sqrt[4]{x-4}=3$, we are dealing with a fourth root. To determine which root to use, look for the index of the root, which is the number outside the radical symbol. In this case, the index is 4, indicating that we are dealing with a fourth root.

Q: Can I use the same steps to solve equations involving different roots?

A: While the steps involved in solving radical equations are similar, the specific steps may vary depending on the root being used. For example, when solving an equation involving a square root, you may need to use the multiplication property of radicals, whereas when solving an equation involving a fourth root, you may need to use the power property of radicals.

Q: What if I have a radical equation with a negative number inside the radical?

A: When dealing with a radical equation that has a negative number inside the radical, you need to be careful when raising both sides of the equation to the power of the index. In some cases, this may result in a negative number on the left-hand side of the equation. To avoid this, you can use the fact that the product of two negative numbers is positive.

Q: Can I use a calculator to solve radical equations?

A: While a calculator can be a useful tool for solving radical equations, it is not always the best approach. In some cases, using a calculator may lead to errors or inaccuracies. Instead, it is often better to use the properties of radicals and the steps involved in solving radical equations to arrive at the solution.

Q: What if I get stuck while solving a radical equation?

A: If you get stuck while solving a radical equation, don't worry! There are several resources available to help you. You can try re-reading the problem, looking for any mistakes or misunderstandings, or seeking help from a teacher or tutor. Additionally, you can try using online resources or video tutorials to help you understand the concept.

In conclusion, solving radical equations can be a challenging but rewarding experience. By understanding the properties of radicals, using the correct steps, and being patient and persistent, you can arrive at the correct solution. We hope that this article has been helpful in addressing some of the most frequently asked questions related to solving radical equations.