IDNLearn.com offers a comprehensive solution for finding accurate answers quickly. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.
Sagot :
Sure, let's go through the solution step-by-step:
### 1. Evaluate
#### (i) \( 2! \)
The factorial of 2, which is \( 2! \), is calculated as:
[tex]\[ 2! = 2 \times 1 = 2 \][/tex]
#### (ii) \( (4.73)^{11} \)
Raising 4.73 to the power of 11 results in:
[tex]\[ (4.73)^{11} \approx 26513834.92968782 \][/tex]
#### (iii) \( 0^1 \)
Any non-negative number to the power of 1 is the number itself. Hence:
[tex]\[ 0^1 = 0 \][/tex]
#### (iv) \( 1^4 \)
Any number raised to any power is 1 if the base is 1. Hence:
[tex]\[ 1^4 = 1 \][/tex]
#### (v) \( (0.25)^1 \)
Any number raised to the power of 1 is the number itself. Hence:
[tex]\[ (0.25)^1 = 0.25 \][/tex]
#### (vi) \( \left(\frac{5}{4}\right)^2 \) and (vii) \( \left(1 \frac{1}{4}\right)^2 \)
Both expressions represent the same value as follows:
[tex]\[ \left(\frac{5}{4}\right)^2 = \left(1.25\right)^2 = 1.5625 \][/tex]
So:
[tex]\[ \left(\frac{5}{4}\right)^2 = 1.5625 \][/tex]
And
[tex]\[ \left(1 \frac{1}{4}\right)^2 = 1.5625 \][/tex]
### 2.
#### (a) Express \(10, 100, 1000, 10000, 100000\) in exponential form
[tex]\[ 10 = 10^1 \][/tex]
[tex]\[ 100 = 10^2 \][/tex]
[tex]\[ 1000 = 10^3 \][/tex]
[tex]\[ 10000 = 10^4 \][/tex]
[tex]\[ 100000 = 10^5 \][/tex]
So the numbers in exponential form are:
[tex]\[ [10^1, 10^2, 10^3, 10^4, 10^5] = [10, 100, 1000, 10000, 100000] \][/tex]
#### (b) Express the following products in simplest exponential form
##### (i) \( 16 \times 64 \)
Both 16 and 64 are powers of 2:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ 64 = 2^6 \][/tex]
Multiplying them together:
[tex]\[ 2^4 \times 2^6 = 2^{4+6} = 2^{10} \][/tex]
So:
[tex]\[ 16 \times 64 = 2^{10} = 1024 \][/tex]
##### (ii) \( 25 \times 125 \)
Both 25 and 125 are powers of 5:
[tex]\[ 25 = 5^2 \][/tex]
[tex]\[ 125 = 5^3 \][/tex]
Multiplying them together:
[tex]\[ 5^2 \times 5^3 = 5^{2+3} = 5^5 \][/tex]
So:
[tex]\[ 25 \times 125 = 5^5 = 3125 \][/tex]
##### (iii) \( 128 + 32 \)
Here, we are dealing with simple addition:
[tex]\[ 128 + 32 = 160 \][/tex]
Thus, the solutions are:
1. Evaluations:
(i) \( 2! = 2 \)
(ii) \( (4.73)^{11} \approx 26513834.92968782 \)
(iii) \( 0^1 = 0 \)
(iv) \( 1^4 = 1 \)
(v) \( (0.25)^1 = 0.25 \)
(vi) \( \left(\frac{5}{4}\right)^2 = 1.5625 \)
(vii) \( \left(1 \frac{1}{4}\right)^2 = 1.5625 \)
2. (a) Exponentials:
[tex]\[ [10^1, 10^2, 10^3, 10^4, 10^5] = [10, 100, 1000, 10000, 100000] \][/tex]
(b) Simplified products:
(i) \( 16 \times 64 = 2^{10} = 1024 \)
(ii) \( 25 \times 125 = 5^5 = 3125 \)
(iii) [tex]\( 128 + 32 = 160 \)[/tex]
### 1. Evaluate
#### (i) \( 2! \)
The factorial of 2, which is \( 2! \), is calculated as:
[tex]\[ 2! = 2 \times 1 = 2 \][/tex]
#### (ii) \( (4.73)^{11} \)
Raising 4.73 to the power of 11 results in:
[tex]\[ (4.73)^{11} \approx 26513834.92968782 \][/tex]
#### (iii) \( 0^1 \)
Any non-negative number to the power of 1 is the number itself. Hence:
[tex]\[ 0^1 = 0 \][/tex]
#### (iv) \( 1^4 \)
Any number raised to any power is 1 if the base is 1. Hence:
[tex]\[ 1^4 = 1 \][/tex]
#### (v) \( (0.25)^1 \)
Any number raised to the power of 1 is the number itself. Hence:
[tex]\[ (0.25)^1 = 0.25 \][/tex]
#### (vi) \( \left(\frac{5}{4}\right)^2 \) and (vii) \( \left(1 \frac{1}{4}\right)^2 \)
Both expressions represent the same value as follows:
[tex]\[ \left(\frac{5}{4}\right)^2 = \left(1.25\right)^2 = 1.5625 \][/tex]
So:
[tex]\[ \left(\frac{5}{4}\right)^2 = 1.5625 \][/tex]
And
[tex]\[ \left(1 \frac{1}{4}\right)^2 = 1.5625 \][/tex]
### 2.
#### (a) Express \(10, 100, 1000, 10000, 100000\) in exponential form
[tex]\[ 10 = 10^1 \][/tex]
[tex]\[ 100 = 10^2 \][/tex]
[tex]\[ 1000 = 10^3 \][/tex]
[tex]\[ 10000 = 10^4 \][/tex]
[tex]\[ 100000 = 10^5 \][/tex]
So the numbers in exponential form are:
[tex]\[ [10^1, 10^2, 10^3, 10^4, 10^5] = [10, 100, 1000, 10000, 100000] \][/tex]
#### (b) Express the following products in simplest exponential form
##### (i) \( 16 \times 64 \)
Both 16 and 64 are powers of 2:
[tex]\[ 16 = 2^4 \][/tex]
[tex]\[ 64 = 2^6 \][/tex]
Multiplying them together:
[tex]\[ 2^4 \times 2^6 = 2^{4+6} = 2^{10} \][/tex]
So:
[tex]\[ 16 \times 64 = 2^{10} = 1024 \][/tex]
##### (ii) \( 25 \times 125 \)
Both 25 and 125 are powers of 5:
[tex]\[ 25 = 5^2 \][/tex]
[tex]\[ 125 = 5^3 \][/tex]
Multiplying them together:
[tex]\[ 5^2 \times 5^3 = 5^{2+3} = 5^5 \][/tex]
So:
[tex]\[ 25 \times 125 = 5^5 = 3125 \][/tex]
##### (iii) \( 128 + 32 \)
Here, we are dealing with simple addition:
[tex]\[ 128 + 32 = 160 \][/tex]
Thus, the solutions are:
1. Evaluations:
(i) \( 2! = 2 \)
(ii) \( (4.73)^{11} \approx 26513834.92968782 \)
(iii) \( 0^1 = 0 \)
(iv) \( 1^4 = 1 \)
(v) \( (0.25)^1 = 0.25 \)
(vi) \( \left(\frac{5}{4}\right)^2 = 1.5625 \)
(vii) \( \left(1 \frac{1}{4}\right)^2 = 1.5625 \)
2. (a) Exponentials:
[tex]\[ [10^1, 10^2, 10^3, 10^4, 10^5] = [10, 100, 1000, 10000, 100000] \][/tex]
(b) Simplified products:
(i) \( 16 \times 64 = 2^{10} = 1024 \)
(ii) \( 25 \times 125 = 5^5 = 3125 \)
(iii) [tex]\( 128 + 32 = 160 \)[/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is committed to providing accurate answers. Thanks for stopping by, and see you next time for more solutions.
