Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Certainly! Let's simplify each of the given expressions step-by-step.
Question 1: Simplify [tex]\( 7^5 \times 7^{-4} \times 7^{-6} \)[/tex]
To simplify this, we'll use the property of exponents which states that when multiplying like bases, you add the exponents: [tex]\(a^m \times a^n = a^{m+n}\)[/tex].
1. Combine the exponents:
[tex]\[ 7^5 \times 7^{-4} \times 7^{-6} = 7^{5 + (-4) + (-6)} \][/tex]
2. Add the exponents:
[tex]\[ 5 + (-4) + (-6) = 5 - 4 - 6 = -5 \][/tex]
3. Express the result:
[tex]\[ 7^{-5} = \frac{1}{7^5} \][/tex]
Evaluating [tex]\(7^{-5}\)[/tex]:
[tex]\[ 7^{-5} \approx 5.9499018266198606 \times 10^{-5} \][/tex]
Question 2: Simplify [tex]\( (-3)^4 \times 7 \)[/tex]
1. Evaluate the first part [tex]\((-3)^4\)[/tex]:
[tex]\[ (-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 81 \][/tex]
2. Multiply by 7:
[tex]\[ 81 \times 7 = 567 \][/tex]
VEL-2:
Part a: Simplify [tex]\( \left(\frac{1}{2}\right)^4 \times \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^6 \)[/tex]
To simplify this, we again use the property of exponents for like bases: [tex]\(a^m \times a^n = a^{m+n}\)[/tex].
1. Combine the exponents:
[tex]\[ \left(\frac{1}{2}\right)^4 \times \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^6 = \left(\frac{1}{2}\right)^{4 + 5 + 6} \][/tex]
2. Add the exponents:
[tex]\[ 4 + 5 + 6 = 15 \][/tex]
3. Express the result:
[tex]\[ \left(\frac{1}{2}\right)^{15} = \frac{1}{2^{15}} \][/tex]
Evaluating [tex]\(\frac{1}{2^{15}}\)[/tex]:
[tex]\[ \frac{1}{2^{15}} \approx 3.0517578125 \times 10^{-5} \][/tex]
Part b: Simplify [tex]\(2^2 \times \frac{3^2}{2^{-2}} \times 3^{-1}\)[/tex]
We'll individually simplify and combine the exponents using properties of exponents.
1. Start by handling the [tex]\( \frac{3^2}{2^{-2}} \)[/tex]:
[tex]\[ \frac{3^2}{2^{-2}} = 3^2 \times 2^2 \][/tex]
Because dividing by [tex]\(2^{-2}\)[/tex] is the same as multiplying by [tex]\(2^2\)[/tex].
2. Now combine all components:
[tex]\[ 2^2 \times (3^2 \times 2^2) \times 3^{-1} \][/tex]
3. Simplify step-by-step:
[tex]\[ 2^2 \times 2^2 \times 3^2 \times 3^{-1} = 2^{2+2} \times 3^{2-1} = 2^4 \times 3^1 \][/tex]
4. Evaluate the simplified expression:
[tex]\[ 2^4 = 16 \quad \text{and} \quad 3^1 = 3 \][/tex]
5. Multiply the results:
[tex]\[ 16 \times 3 = 48 \][/tex]
Thus, the simplified results for each part are:
1. [tex]\( 7^5 \times 7^{-4} \times 7^{-6} \approx 5.9499018266198606 \times 10^{-5} \)[/tex]
2. [tex]\( (-3)^4 \times 7 = 567 \)[/tex]
3. a) [tex]\( \left(\frac{1}{2}\right)^4 \times \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^6 \approx 3.0517578125 \times 10^{-5} \)[/tex]
b) [tex]\( 2^2 \times \frac{3^2}{2^{-2}} \times 3^{-1} = 48\)[/tex]
These values are expressed in their simplest forms and evaluated accurately based on the operations involved.
Question 1: Simplify [tex]\( 7^5 \times 7^{-4} \times 7^{-6} \)[/tex]
To simplify this, we'll use the property of exponents which states that when multiplying like bases, you add the exponents: [tex]\(a^m \times a^n = a^{m+n}\)[/tex].
1. Combine the exponents:
[tex]\[ 7^5 \times 7^{-4} \times 7^{-6} = 7^{5 + (-4) + (-6)} \][/tex]
2. Add the exponents:
[tex]\[ 5 + (-4) + (-6) = 5 - 4 - 6 = -5 \][/tex]
3. Express the result:
[tex]\[ 7^{-5} = \frac{1}{7^5} \][/tex]
Evaluating [tex]\(7^{-5}\)[/tex]:
[tex]\[ 7^{-5} \approx 5.9499018266198606 \times 10^{-5} \][/tex]
Question 2: Simplify [tex]\( (-3)^4 \times 7 \)[/tex]
1. Evaluate the first part [tex]\((-3)^4\)[/tex]:
[tex]\[ (-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 81 \][/tex]
2. Multiply by 7:
[tex]\[ 81 \times 7 = 567 \][/tex]
VEL-2:
Part a: Simplify [tex]\( \left(\frac{1}{2}\right)^4 \times \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^6 \)[/tex]
To simplify this, we again use the property of exponents for like bases: [tex]\(a^m \times a^n = a^{m+n}\)[/tex].
1. Combine the exponents:
[tex]\[ \left(\frac{1}{2}\right)^4 \times \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^6 = \left(\frac{1}{2}\right)^{4 + 5 + 6} \][/tex]
2. Add the exponents:
[tex]\[ 4 + 5 + 6 = 15 \][/tex]
3. Express the result:
[tex]\[ \left(\frac{1}{2}\right)^{15} = \frac{1}{2^{15}} \][/tex]
Evaluating [tex]\(\frac{1}{2^{15}}\)[/tex]:
[tex]\[ \frac{1}{2^{15}} \approx 3.0517578125 \times 10^{-5} \][/tex]
Part b: Simplify [tex]\(2^2 \times \frac{3^2}{2^{-2}} \times 3^{-1}\)[/tex]
We'll individually simplify and combine the exponents using properties of exponents.
1. Start by handling the [tex]\( \frac{3^2}{2^{-2}} \)[/tex]:
[tex]\[ \frac{3^2}{2^{-2}} = 3^2 \times 2^2 \][/tex]
Because dividing by [tex]\(2^{-2}\)[/tex] is the same as multiplying by [tex]\(2^2\)[/tex].
2. Now combine all components:
[tex]\[ 2^2 \times (3^2 \times 2^2) \times 3^{-1} \][/tex]
3. Simplify step-by-step:
[tex]\[ 2^2 \times 2^2 \times 3^2 \times 3^{-1} = 2^{2+2} \times 3^{2-1} = 2^4 \times 3^1 \][/tex]
4. Evaluate the simplified expression:
[tex]\[ 2^4 = 16 \quad \text{and} \quad 3^1 = 3 \][/tex]
5. Multiply the results:
[tex]\[ 16 \times 3 = 48 \][/tex]
Thus, the simplified results for each part are:
1. [tex]\( 7^5 \times 7^{-4} \times 7^{-6} \approx 5.9499018266198606 \times 10^{-5} \)[/tex]
2. [tex]\( (-3)^4 \times 7 = 567 \)[/tex]
3. a) [tex]\( \left(\frac{1}{2}\right)^4 \times \left(\frac{1}{2}\right)^5 \times \left(\frac{1}{2}\right)^6 \approx 3.0517578125 \times 10^{-5} \)[/tex]
b) [tex]\( 2^2 \times \frac{3^2}{2^{-2}} \times 3^{-1} = 48\)[/tex]
These values are expressed in their simplest forms and evaluated accurately based on the operations involved.
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
