Let's solve each part step by step.
### Part (a)
Given:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = 2 \)[/tex]
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 4 \)[/tex]
We need to find the value of:
[tex]\[ \frac{a^m \times b^n \times c^n}{m^2 \times m^5 \times (m \cdot n)^6} \][/tex]
First, calculate the numerator:
[tex]\[ a^m \times b^n \times c^n \][/tex]
Substitute the given values:
[tex]\[ 3^3 \times 1^4 \times 2^4 \][/tex]
Evaluate the powers:
[tex]\[ 27 \times 1 \times 16 \][/tex]
Now, multiply these results:
[tex]\[ 27 \times 16 = 432 \][/tex]
Next, calculate the denominator:
[tex]\[ m^2 \times m^5 \times (m \cdot n)^6 \][/tex]
Substitute the given values:
[tex]\[ 3^2 \times 3^5 \times (3 \times 4)^6 \][/tex]
Evaluate the powers:
[tex]\[ 9 \times 243 \times (12)^6 \][/tex]
Calculate [tex]\( (12)^6 \)[/tex]:
[tex]\[ 12^6 = 2985984 \][/tex]
Now, multiply these results:
[tex]\[ 9 \times 243 \times 2985984 = 6530347008 \][/tex]
Finally, divide the numerator by the denominator:
[tex]\[ \frac{432}{6530347008} \approx 6.61526867516808 \times 10^{-8} \][/tex]
So the value of part (a) is approximately:
[tex]\[ 6.61526867516808 \times 10^{-8} \][/tex]
### Part (b)
Given the same values, we now need to find:
[tex]\[ \frac{(a+b-c)^{m+n}}{(n-m)^{-b+c}} \][/tex]
Substitute the given values:
[tex]\[ \frac{(3 + 1 - 2)^{3+4}}{(4 - 3)^{-1+2}} \][/tex]
Simplify inside the parentheses:
[tex]\[ \frac{(2)^7}{(1)^1} \][/tex]
Evaluate the powers:
[tex]\[ \frac{128}{1} \][/tex]
So the value of part (b) is:
[tex]\[ 128 \][/tex]
In summary:
- Part (a): [tex]\( 6.61526867516808 \times 10^{-8} \)[/tex]
- Part (b): [tex]\( 128 \)[/tex]
