IDNLearn.com provides a seamless experience for finding and sharing answers. Explore thousands of verified answers from experts and find the solutions you need, no matter the topic.
Sagot :
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]
### 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]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.
