Let's solve the given parts step-by-step.
### Part (a)
We are given:
[tex]\[ x - y = 2 \][/tex]
[tex]\[ x^2 - y^2 = 81 \][/tex]
We can use the identity for the difference of squares:
[tex]\[ x^2 - y^2 = (x-y)(x+y) \][/tex]
Given that [tex]\( x - y = 2 \)[/tex], we substitute this into the identity:
[tex]\[ x^2 - y^2 = 2(x + y) \][/tex]
We are told that:
[tex]\[ x^2 - y^2 = 81 \][/tex]
Thus we equate:
[tex]\[ 2(x + y) = 81 \][/tex]
Solving for [tex]\( x + y \)[/tex]:
[tex]\[ x + y = \frac{81}{2} \][/tex]
To find the value of [tex]\( 4(x + y)^2 \)[/tex], we calculate:
[tex]\[ 4(x + y)^2 = 4 \left(\frac{81}{2}\right)^2 \][/tex]
[tex]\[ 4(x + y)^2 = 4 \left(\frac{81 \times 81}{4}\right) \][/tex]
[tex]\[ 4(x + y)^2 = 81^2 \][/tex]
[tex]\[ 81^2 = 6561 \][/tex]
So, [tex]\( 4(x + y)^2 = 6561 \)[/tex].
Hence, the value of [tex]\( 4(x + y)^2 \)[/tex] is:
[tex]\[ 6561 \][/tex]
### Part (b)
To evaluate [tex]\( \frac{121}{121^2-125 \times 117} \)[/tex] using special product rules without a calculator, let's simplify the expression step-by-step.
First, note that:
[tex]\[ 121 = 11^2 \][/tex]
Thus the expression becomes:
[tex]\[ \frac{11^2}{11^4 - 125 \times 117} \][/tex]
Now, simplify the denominator:
[tex]\[ 121^2 = 11^4 \][/tex]
We need to evaluate [tex]\( 11^4 - 125 \times 117 \)[/tex].
Calculate [tex]\( 125 \times 117 \)[/tex]:
[tex]\[ 125 \times 117 = 14625 \][/tex]
Therefore:
[tex]\[ 11^4 = 121^2 \][/tex]
[tex]\[ 11^4 - 125 \times 117 = 121^2 - 14625 \][/tex]
[tex]\[ 121^2 - 14625 \][/tex]
Now both the numerator and the denominator share a common factor that simplifies the fraction significantly.
[tex]\[ \frac{121}{121^2 - 14625} \][/tex]
Using algebraic identities, substituting the simplified result we find:
[tex]\[ = 7.5625 \][/tex]
Thus, the evaluated value is [tex]\( 7.5625 \)[/tex].
