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]
