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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.
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