Certainly! Let's go through each of the given expressions and simplify them step-by-step:
### Expression 1:
[tex]\[ \frac{2^3 \cdot 2^{-2}}{2^5} \][/tex]
1. Simplify the numerator:
- Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
2^3 \cdot 2^{-2} = 2^{3 + (-2)} = 2^{1} = 2
\][/tex]
2. Combine the simplified numerator with the denominator:
- Now, the expression is:
[tex]\[
\frac{2^1}{2^5} = 2^{1-5} = 2^{-4}
\][/tex]
3. Simplify the exponent:
- A negative exponent indicates a reciprocal:
[tex]\[
2^{-4} = \frac{1}{2^4} = \frac{1}{16}
\][/tex]
So, the simplified form of the first expression is:
[tex]\[ \frac{2^3 \cdot 2^{-2}}{2^5} = \frac{1}{16} \][/tex]
### Expression 2:
[tex]\[ \left(a^3\right)^4 \cdot b^2 \cdot b^{-3} \][/tex]
1. Simplify the part [tex]\(\left(a^3\right)^4\)[/tex]:
- Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
\left(a^3\right)^4 = a^{3 \cdot 4} = a^{12}
\][/tex]
2. Simplify the part [tex]\( b^2 \cdot b^{-3} \)[/tex]:
- Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
b^2 \cdot b^{-3} = b^{2 + (-3)} = b^{-1}
\][/tex]
3. Express [tex]\( b^{-1} \)[/tex] as a fraction:
- A negative exponent indicates a reciprocal:
[tex]\[
b^{-1} = \frac{1}{b}
\][/tex]
4. Combine everything:
[tex]\[
a^{12} \cdot b^{-1} = \frac{a^{12}}{b}
\][/tex]
So, the simplified form of the second expression is:
[tex]\[ \left(a^3\right)^4 \cdot b^2 \cdot b^{-3} = \frac{a^{12}}{b} \][/tex]
### Expression 3:
[tex]\[ \frac{P^2 \cdot (Q^2)^3}{P^1} \][/tex]
1. Simplify the part [tex]\((Q^2)^3\)[/tex]:
- Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(Q^2)^3 = Q^{2 \cdot 3} = Q^6
\][/tex]
2. Combine the simplified parts [tex]\(P^2\)[/tex] and [tex]\(Q^6\)[/tex]:
- Now, the expression is:
[tex]\[
\frac{P^2 \cdot Q^6}{P}
\][/tex]
3. Simplify by combining [tex]\(P^2\)[/tex] and [tex]\(P^1\)[/tex]:
- Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{P^2}{P^1} = P^{2-1} = P
\][/tex]
4. Combine everything:
[tex]\[
P \cdot Q^6
\][/tex]
So, the simplified form of the third expression is:
[tex]\[ \frac{P^2 \cdot (Q^2)^3}{P^1} = P \cdot Q^6 \][/tex]
### Summary:
1. [tex]\(\frac{2^3 \cdot 2^{-2}}{2^5} = \frac{1}{16}\)[/tex]
2. [tex]\(\left(a^3\right)^4 \cdot b^2 \cdot b^{-3} = \frac{a^{12}}{b}\)[/tex]
3. [tex]\(\frac{P^2 \cdot (Q^2)^3}{P^1} = P \cdot Q^6\)[/tex]
