`(sqrt8)^6xx(64)^3div(8)^?=(8)^5`

Options

A. 3
B. 1
C. 5
D. None of these

Answer

To solve the equation \((\sqrt{8})^6 \times (64)^3 \div (8)^{?} = (8)^5\), we will follow these steps:

### Step 1: Rewrite the terms in the equation using the same base
First, we need to express all the terms in the equation with base 8.

1. \(\sqrt{8}\) can be rewritten as \(8^{1/2}\).
2. Therefore, \((\sqrt{8})^6 = (8^{1/2})^6 = 8^{6/2} = 8^3\).
3. Next, \(64\) can be rewritten as \(8^2\) because \(64 = 8^2\).
4. Thus, \((64)^3 = (8^2)^3 = 8^{2 \times 3} = 8^6\).

Now we can rewrite the entire equation:
\[ (8^3) \times (8^6) \div (8)^{?} = (8)^5 \]

### Step 2: Combine the powers of 8
Using the property of exponents that states \(a^m \times a^n = a^{m+n}\), we can combine \(8^3\) and \(8^6\):
\[ 8^{3 + 6} \div (8)^{?} = 8^5 \]
This simplifies to:
\[ 8^9 \div (8)^{?} = 8^5 \]

### Step 3: Apply the division of powers
Using the property of exponents that states \(a^m \div a^n = a^{m-n}\), we can simplify the left side:
\[ 8^{9 - ?} = 8^5 \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 9 - ? = 5 \]

### Step 5: Solve for the question mark
Now, we can solve for \(?\):
\[ 9 - 5 = ? \]
\[ ? = 4 \]

### Final Answer
The value of \(?\) is \(4\).

---