Answer :
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]
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]
