`(sqrt(9))^(2)xx(sqrt(81))^(5)div(27)^(2)=9^(?)`

Options

A. 3
B. 5
C. 6
D. 8

Explanation

To solve the equation \((\sqrt{9})^{2} \times (\sqrt{81})^{5} \div (27)^{2} = 9^{?}\), we will simplify each part step by step.

### Step 1: Simplify \((\sqrt{9})^{2}\)
\[ \sqrt{9} = 3 \]
Thus,
\[ (\sqrt{9})^{2} = 3^{2} = 9 \]

**Hint:** Remember that the square root of a number squared returns the original number.

### Step 2: Simplify \((\sqrt{81})^{5}\)
\[ \sqrt{81} = 9 \]
Thus,
\[ (\sqrt{81})^{5} = 9^{5} \]

**Hint:** The square root of a perfect square gives you the base of that square.

### Step 3: Simplify \((27)^{2}\)
\[ 27 = 3^{3} \quad \text{(since } 3 \times 3 \times 3 = 27\text{)} \]
Thus,
\[ (27)^{2} = (3^{3})^{2} = 3^{6} \]

**Hint:** When raising a power to another power, multiply the exponents.

### Step 4: Combine the terms
Now we can rewrite the original expression using our simplifications:
\[ 9 \times 9^{5} \div 3^{6} \]
This can be expressed as:
\[ 9^{1} \times 9^{5} \div 3^{6} \]

**Hint:** When multiplying like bases, add the exponents.

### Step 5: Combine the powers of 9
\[ 9^{1 + 5} = 9^{6} \]

**Hint:** Adding exponents is a key property of exponents when multiplying.

### Step 6: Convert \(9^{6}\) to base 3
Since \(9 = 3^{2}\), we can write:
\[ 9^{6} = (3^{2})^{6} = 3^{12} \]

**Hint:** Changing the base involves multiplying the exponents.

### Step 7: Set the equation equal to \(9^{?}\)
Now we have:
\[ 3^{12} = 9^{?} \]
Since \(9 = 3^{2}\), we can express \(9^{?}\) as:
\[ 9^{?} = (3^{2})^{?} = 3^{2?} \]

### Step 8: Equate the exponents
Setting the exponents equal gives us:
\[ 12 = 2? \]

### Step 9: Solve for \(?\)
Dividing both sides by 2:
\[ ? = \frac{12}{2} = 6 \]

Thus, the final answer is:
\[ 9^{6} \]

### Final Answer:
\[ ? = 6 \]