Sure, let's solve each expression step by step using the rules of exponents.
1. First Expression: [tex]\(\frac{2^3 \cdot 2^{-2}}{2^5}\)[/tex]
- Simplify the numerator using the rule [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]\[
2^3 \cdot 2^{-2} = 2^{3 + (-2)} = 2^1
\][/tex]
- Now, simplify the whole fraction using the rule [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[
\frac{2^1}{2^5} = 2^{1-5} = 2^{-4}
\][/tex]
- Convert the exponent to a positive by using [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]:
[tex]\[
2^{-4} = \frac{1}{2^4} = \frac{1}{16} = 0.0625
\][/tex]
So, the result of the first expression is [tex]\( 0.0625 \)[/tex].
2. Second Expression: [tex]\(\left(a^3\right)^4 \cdot b^2 \cdot b^{-3}\)[/tex]
- Simplify the [tex]\( a \)[/tex] term using the rule [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex]:
[tex]\[
\left(a^3\right)^4 = a^{3 \cdot 4} = a^{12}
\][/tex]
- Simplify the [tex]\( b \)[/tex] terms using the rule [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]:
[tex]\[
b^2 \cdot b^{-3} = b^{2 + (-3)} = b^{-1}
\][/tex]
So, the result of the second expression is [tex]\( a^{12} \cdot b^{-1} \)[/tex] or simply the exponents as [tex]\((12, -1)\)[/tex].
3. Third Expression: [tex]\(\frac{P^2 \cdot \left(Q^2\right)^3}{P^1}\)[/tex]
- Simplify the [tex]\( Q \)[/tex] term using the rule [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex]:
[tex]\[
\left(Q^2\right)^3 = Q^{2 \cdot 3} = Q^6
\][/tex]
- Simplify the numerator combining all the [tex]\( P \)[/tex] terms:
[tex]\[
P^2 \cdot Q^6
\][/tex]
- Now, simplify the whole fraction using the rule [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[
\frac{P^2 \cdot Q^6}{P^1} = P^{2-1} \cdot Q^6 = P^1 \cdot Q^6 = P \cdot Q^6
\][/tex]
So, the result of the third expression is [tex]\( P^1 \cdot Q^6 \)[/tex] or simply the exponents as [tex]\((1, 6)\)[/tex].
Thus, the detailed step-by-step solutions for the given expressions are:
1. [tex]\(\frac{2^3 \cdot 2^{-2}}{2^5} = 0.0625\)[/tex]
2. [tex]\(\left(a^3\right)^4 \cdot b^2 \cdot b^{-3} = (12, -1)\)[/tex]
3. [tex]\(\frac{P^2 \cdot \left(Q^2\right)^3}{P^1} = (1, 6)\)[/tex]
