To simplify the expression [tex]\( 27^{-2/3} \div 9^{1/2} \div 3^{-3/2} \)[/tex], we will follow these steps:
1. Express each base as a power of 3:
- Recognize that [tex]\( 27 = 3^3 \)[/tex] and [tex]\( 9 = 3^2 \)[/tex]. Rewrite the expression using these identities:
[tex]\[
27^{-2/3} = (3^3)^{-2/3}
\][/tex]
[tex]\[
9^{1/2} = (3^2)^{1/2}
\][/tex]
[tex]\[
3^{-3/2} \text{ remains as it is.}
\][/tex]
2. Simplify each term:
- Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], simplify each term:
[tex]\[
(3^3)^{-2/3} = 3^{3 \cdot (-2/3)} = 3^{-2}
\][/tex]
[tex]\[
(3^2)^{1/2} = 3^{2 \cdot (1/2)} = 3^1 = 3
\][/tex]
[tex]\[
3^{-3/2} \text{ remains as it is.}
\][/tex]
So, the expression now looks like this:
[tex]\[
3^{-2} \div 3^1 \div 3^{-3/2}
\][/tex]
3. Combine the terms using the properties of exponents:
- Use the rule for division of exponents with the same base, [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], to combine the terms. We do this in steps:
First, deal with [tex]\( 3^{-2} \div 3^1 \)[/tex]:
[tex]\[
3^{-2} \div 3^1 = 3^{-2 - 1} = 3^{-3}
\][/tex]
Then, combine this with [tex]\( 3^{-3/2} \)[/tex]:
[tex]\[
3^{-3} \div 3^{-3/2} = 3^{-3 - (-3/2)}
\][/tex]
Simplify the exponent:
[tex]\[
-3 - (-3/2) = -3 + 3/2 = -3 + 1.5 = -4.5
\][/tex]
So, we have:
[tex]\[
3^{-4.5}
\][/tex]
4. Final result:
[tex]\[
3^{-4.5}
\][/tex]
Converting the final expression back into a numerical result, we get:
[tex]\[
3^{-4.5} \approx 0.007127781101106491
\][/tex]
Thus, the simplified form of [tex]\( 27^{-2/3} \div 9^{1/2} \div 3^{-3/2} \)[/tex] is approximately:
[tex]\[
\boxed{0.007127781101106491}
\][/tex]
