Certainly! Let's solve the expression [tex]\( 9^3 \times 3^{-3} \times 27^2 \)[/tex] step-by-step.
### Step 1: Express each term with a common base
First, rewrite all terms so they share a common base of 3:
- [tex]\( 9 \)[/tex] is a power of 3. Specifically, [tex]\( 9 = 3^2 \)[/tex].
- [tex]\( 27 \)[/tex] is also a power of 3. Specifically, [tex]\( 27 = 3^3 \)[/tex].
Now, express [tex]\( 9^3 \)[/tex] and [tex]\( 27^2 \)[/tex] in terms of base 3:
[tex]\[ 9^3 = (3^2)^3 = 3^{2 \times 3} = 3^6 \][/tex]
[tex]\[ 3^{-3} \text{ remains the same as } 3^{-3} \][/tex]
[tex]\[ 27^2 = (3^3)^2 = 3^{3 \times 2} = 3^6 \][/tex]
### Step 2: Combine the exponents
Use the properties of exponents to combine the terms:
[tex]\[ 9^3 \times 3^{-3} \times 27^2 = 3^6 \times 3^{-3} \times 3^6 \][/tex]
When multiplying exponential terms with the same base, you add the exponents:
[tex]\[ 3^6 \times 3^{-3} \times 3^6 = 3^{6 + (-3) + 6} \][/tex]
### Step 3: Simplify the exponent sum
Now, calculate the sum of the exponents:
[tex]\[ 6 + (-3) + 6 = 9 \][/tex]
So, the expression simplifies to:
[tex]\[ 3^9 \][/tex]
### Step 4: Calculate the final result
The value of [tex]\( 3^9 \)[/tex] is:
[tex]\[ 3^9 = 19683 \][/tex]
Therefore, the value of the given expression [tex]\( 9^3 \times 3^{-3} \times 27^2 \)[/tex] is:
[tex]\[
\boxed{19683}
\][/tex]
Summary:
- The expression [tex]\( 9^3 \times 3^{-3} \times 27^2 \)[/tex] simplifies to [tex]\( 3^9 \)[/tex], which equals [tex]\( 19683 \)[/tex].
