Root of a Root: Solving the equation

To solve this problem, let's proceed with the following steps:

• Step 1: Start with the given equation:
  $\frac{\sqrt{98}}{\sqrt{x}} = 7$.
• Step 2: Apply the square root property to combine the fraction:
  $\sqrt{\frac{98}{x}} = 7$.
• Step 3: Square both sides to eliminate the square root:
  $\frac{98}{x} = 49$.
• Step 4: Solve for $x$ by multiplying both sides by $x$:
  $98 = 49x$.
• Step 5: Isolate $x$ by dividing both sides by 49:
  $x = \frac{98}{49}$.
• Step 6: Simplify the fraction:
  $x = \frac{98}{49} = 2$.

Therefore, the solution to the problem is $x = 2$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify the left side of the equation, $\sqrt{\sqrt{81}}$.
• Step 2: Simplify the right side of the equation, $\sqrt[3]{\sqrt{x^6}}$.
• Step 3: Equate the simplified expressions and solve for $x$.

Now, let's simplify each side:

Step 1: Simplify $\sqrt{\sqrt{81}}$.

First, evaluate $\sqrt{81}$, which is $9$, since $9^2 = 81$.
Then, evaluate $\sqrt{9}$, which is $3$, since $3^2 = 9$.
So, $\sqrt{\sqrt{81}} = 3$.

Step 2: Simplify $\sqrt[3]{\sqrt{x^6}}$.

Express $\sqrt{x^6}$ as $(x^6)^{1/2} = x^{6/2} = x^3$.
Express $\sqrt[3]{x^3}$ as $(x^3)^{1/3} = x^{3/3} = x^1 = x$.
So, $\sqrt[3]{\sqrt{x^6}} = x$.

Step 3: Set the simplified expressions equal.

We have simplified both sides of the equation to get $3 = x$.
Therefore, the solution to the problem is $x = 3$.

Hence, the correct answer is $x = 3$.

Therefore, the correct choice is:

Choice 2: $x = 3$.

To solve this equation, we will follow these steps:

• Step 1: Simplify the left side of the equation $\sqrt{144}$.
• Step 2: Simplify the right side of the equation $\sqrt[3]{\sqrt[5]{x^{30}}}$.
• Step 3: Equate the simplified expressions and solve for $x$.

Let us go through these steps:

Step 1: Simplify the left side:

The left side of the equation is $\sqrt{144}$, which simplifies to $12$, because $\sqrt{144} = 12$.

Step 2: Simplify the right side:

The expression on the right is $\sqrt[3]{\sqrt[5]{x^{30}}}$. Let's simplify it step by step:

• First, simplify $\sqrt[5]{x^{30}}$:
  - Using the rule $\sqrt[n]{a^m} = a^{m/n}$, we have $\sqrt[5]{x^{30}} = x^{30/5} = x^6$.
• Next, simplify $\sqrt[3]{x^6}$:
  - Again using the same rule, $\sqrt[3]{x^6} = x^{6/3} = x^2$.

Step 3: Equate and solve:

From the previous steps, we get:

$12 = x^2$

Therefore, the solution to the equation is:
$x^2 = 12$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify $\sqrt{\sqrt{16}}$.
• Step 2: Simplify $\sqrt{25}$.
• Step 3: Calculate the right-hand side.
• Step 4: Solve $\sqrt{x^6} =$ computed right-hand side.

Now, let's work through each step:
Step 1: Simplify $\sqrt{\sqrt{16}}$.
We know that $\sqrt{16} = 4$, and $\sqrt{4} = 2$. So, $\sqrt{\sqrt{16}} = 2$.

Step 2: Simplify $\sqrt{25}$.
We know that $\sqrt{25} = 5$.

Step 3: Calculate the entire right-hand side.
We have $\sqrt{\sqrt{16}} \cdot \sqrt{25} = 2 \cdot 5 = 10$.

Step 4: Solve $\sqrt{x^6} = 10$.
Rewrite the left side as $(x^6)^{1/2} = x^{6/2} = x^3$.
Thus, $x^3 = 10$.

Therefore, the solution to the problem is $x^3 = 10$.

To solve this problem, we'll begin by simplifying both sides of the equation:

• Simplifying the left-hand side:

$\sqrt[5]{\sqrt{x^{10}}}$ can be rewritten using properties of exponents and roots. The inner square root is $(x^{10})^{1/2} = x^{10 \cdot \frac{1}{2}} = x^{5}$.

Then, take the fifth root: $(x^{5})^{\frac{1}{5}} = x^{5 \cdot \frac{1}{5}} = x^{1} = x$.
Thus, the left-hand side simplifies to $x$.

• Simplifying the right-hand side:

$\sqrt{\sqrt{81}}$ simplifies as follows: First, find $\sqrt{81} = 9$.
Then, compute $\sqrt{9} = 3$.

So, the equation reduces to $x = 3$.

Therefore, the solution to the problem is $\boxed{x = 3}$.

To solve the problem $\sqrt{\frac{16}{\sqrt[3]{64}}} = \sqrt{x^2}$, we proceed step-by-step as follows:

• First, we simplify $\sqrt[3]{64}$. Since $64 = 4^3$, it follows that $\sqrt[3]{64} = 4$.
• Next, simplify the expression $\frac{16}{\sqrt[3]{64}}$:
  $\frac{16}{4} = 4$.
• Now, take the square root of this simplified value. Thus, $\sqrt{4} = 2$.
• The equation simplifies to: $\sqrt{x^2} = 2$. Since $\sqrt{x^2} = |x|$, we have $|x| = 2$.
• This implies $x = 2$ or $x = -2$.
• However, choices include only positive solutions, and thus $x = 2$.

Therefore, the solution to the problem is $x = 2$.