At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
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.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
