Solving The Radical Equation: \sqrt[8]{2x-7}=3

When we talk about solving radical equations, we're essentially looking for the value of the variable that makes the equation true. In this particular case, we have the equation ${\sqrt[8]{2x-7}=3}$. This is an eighth-root equation, meaning we have a variable inside a root raised to the power of 8. The primary goal when tackling such problems is to isolate the variable, 'x', and find its numerical value. This process involves a series of algebraic steps, each designed to simplify the equation and move us closer to our objective. We need to be particularly careful with radical equations because raising both sides to a power can sometimes introduce extraneous solutions – solutions that satisfy the modified equation but not the original one. Therefore, it's always a good practice to check our final answer by substituting it back into the initial equation to ensure it holds true. The journey to solving ${\sqrt[8]{2x-7}=3}$ will involve understanding the properties of exponents and roots, and applying them methodically. We'll begin by eliminating the radical, then proceed to isolate 'x', and finally, we'll confirm our solution. This approach not only helps us find the correct answer but also deepens our understanding of how these mathematical concepts interact.

Step 1: Isolating the Radical

The first crucial step in solving the radical equation ${\sqrt[8]{2x-7}=3}$ is to ensure that the radical term is isolated on one side of the equation. In this specific equation, the radical term, ${\sqrt[8]{2x-7}}$, is already beautifully isolated on the left-hand side. This means we don't need to perform any addition, subtraction, multiplication, or division to move other terms away from it. This simplification at the outset makes the subsequent steps more straightforward. If the equation had been, for instance, ${\sqrt[8]{2x-7} + 5 = 8}$, our first action would have been to subtract 5 from both sides to get ${\sqrt[8]{2x-7} = 3}$. But here, we're already starting from a point of advantage. The presence of the eighth root signifies that we are looking for a number which, when multiplied by itself eight times, equals ${2x-7}$. The number 3 on the right side of the equation is the result of this operation. So, with the radical term on its own, we can now focus on eliminating it to get to the expression inside, ${2x-7}$.

Step 2: Eliminating the Radical

To eliminate the eighth root in the equation ${\sqrt[8]{2x-7}=3}$, we need to perform the inverse operation of taking the eighth root. The inverse operation of taking an nth root is raising to the nth power. Since we have an eighth root, we will raise both sides of the equation to the power of 8. This is a fundamental property of exponents and radicals: ${(\sqrt[n]{a})^n = a}$ and ${(a^n)^{1/n} = a}$ (under certain conditions). Applying this to our equation, we get:

${(\sqrt[8]{2x-7})^8 = 3^8}$

On the left side, the eighth power and the eighth root cancel each other out, leaving us with just the expression inside the radical:

${2x-7}$

On the right side, we need to calculate ${3^8}$. This means multiplying 3 by itself eight times:

${3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}$

Let's break this down:

${3^2 = 9}$

${3^4 = 3^2 \times 3^2 = 9 \times 9 = 81}$

${3^8 = 3^4 \times 3^4 = 81 \times 81}$

Now, let's calculate ${81 \times 81}$:

81 x 81

81 (1 \times 81) 6480 (80 \times 81)

6561

So, ${3^8 = 6561}$.

Our equation now simplifies to:

${2x-7 = 6561}$

This step has successfully removed the radical and presented us with a much simpler linear equation to solve for 'x'. Remember, the goal is to isolate 'x', and this has brought us one step closer.

Step 3: Isolating the Variable 'x'

Now that we have the simplified equation ${2x-7 = 6561}$, the next logical step in solving for the variable 'x' is to isolate it. This is a standard procedure for linear equations. We want to get 'x' all by itself on one side of the equation. Currently, 'x' is being multiplied by 2, and then 7 is being subtracted from the result. To reverse these operations and isolate 'x', we'll perform the inverse operations in the reverse order of operations (PEMDAS in reverse - SADMEP). First, we need to undo the subtraction of 7. We do this by adding 7 to both sides of the equation:

${2x - 7 + 7 = 6561 + 7}$

This simplifies to:

${2x = 6568}$

Now, 'x' is being multiplied by 2. To undo this multiplication, we will divide both sides of the equation by 2:

${\frac{2x}{2} = \frac{6568}{2}}$

This gives us:

${x = 3284}$

So, we have found a potential solution for our radical equation. The value of 'x' is 3284. However, as mentioned earlier, it's crucial to verify this solution in the original equation to ensure it's not an extraneous root. This check is particularly important for radical equations, though in this specific case, since we raised both sides to an even power (8), it's good practice to confirm.

Step 4: Checking the Solution

The final and arguably most important step in solving the radical equation ${\sqrt[8]{2x-7}=3}$ is to check our calculated value of x. We found that ${x = 3284}$. To verify this, we substitute this value back into the original equation:

${\sqrt[8]{2x-7} = 3}$

Substitute ${x = 3284}$:

${\sqrt[8]{2(3284)-7}}$

First, calculate the expression inside the radical:

${2 \times 3284 = 6568}$

Now, subtract 7:

${6568 - 7 = 6561}$

So, the expression inside the radical becomes 6561. Our equation now looks like:

${\sqrt[8]{6561}}$

We need to determine if the eighth root of 6561 is indeed 3. We already calculated in Step 2 that ${3^8 = 6561}$. Therefore, the eighth root of 6561 is 3:

${\sqrt[8]{6561} = 3}$

Since the left side of the equation equals the right side (3 = 3), our solution ${x = 3284}$ is correct and is not an extraneous root. This confirms that we have successfully solved the radical equation.

Conclusion

Solving the radical equation ${\sqrt[8]{2x-7}=3}$ involved a systematic approach of isolating the radical, raising both sides to the appropriate power to eliminate the radical, and then solving the resulting linear equation. The final verification step confirmed that our solution, ${x=3284}$, is indeed accurate. This process highlights the importance of understanding inverse operations and the potential for extraneous solutions when dealing with even-powered roots. For further exploration into solving various types of equations, you can visit Khan Academy's mathematics section for comprehensive resources and practice problems.