`81^(2.5)xx9^(4.5)/3^(4.8)=9^(?)`

Options

A. 7.1
B. 9.4
C. 4.7
D. 4.5

Answer

To solve the equation \( \frac{81^{2.5} \times 9^{4.5}}{3^{4.8}} = 9^{?} \), we will first express all the terms in the equation in terms of base 3.

### Step 1: Convert the bases
We know that:
- \( 81 = 3^4 \)
- \( 9 = 3^2 \)

Now, we can rewrite the equation:

\[ 81^{2.5} = (3^4)^{2.5} = 3^{4 \times 2.5} = 3^{10} \]
\[ 9^{4.5} = (3^2)^{4.5} = 3^{2 \times 4.5} = 3^{9} \]
\[ 3^{4.8} = 3^{4.8} \]

### Step 2: Substitute back into the equation
Now substituting these values back into the equation gives us:

\[ \frac{3^{10} \times 3^{9}}{3^{4.8}} = 9^{?} \]

### Step 3: Simplify the left side
Using the property of exponents \( a^m \times a^n = a^{m+n} \):

\[ 3^{10 + 9 - 4.8} = 3^{18 - 4.8} = 3^{13.2} \]

### Step 4: Convert the right side
Now we need to express \( 9^{?} \) in terms of base 3:

\[ 9^{?} = (3^2)^{?} = 3^{2?} \]

### Step 5: Set the exponents equal
Now we can set the exponents equal to each other:

\[ 13.2 = 2? \]

### Step 6: Solve for ?
Now, divide both sides by 2:

\[ ? = \frac{13.2}{2} = 6.6 \]

### Final Answer
Thus, the value of \( ? \) is \( 6.6 \).

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