Let's break down the given expression step by step and evaluate it for [tex]\( x = 4 \)[/tex] and [tex]\( y = -3 \)[/tex].
Given expression:
[tex]\[
\frac{64}{x^2 + 2y^2 - 2}
\][/tex]
First, we need to determine the values of [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex]:
[tex]\[
x^2 = 4^2 = 16
\][/tex]
[tex]\[
y^2 = (-3)^2 = 9
\][/tex]
Next, we substitute [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex] into the denominator of the given expression:
[tex]\[
x^2 + 2y^2 - 2 = 16 + 2(9) - 2
\][/tex]
Calculate [tex]\( 2(9) \)[/tex]:
[tex]\[
2 \cdot 9 = 18
\][/tex]
Now, adding and subtracting these values:
[tex]\[
16 + 18 - 2 = 32
\][/tex]
Therefore, the denominator of the expression is 32.
Now, substitute this value back into the expression:
[tex]\[
\frac{64}{32}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{64}{32} = 2
\][/tex]
Hence, the value of the expression when [tex]\( x = 4 \)[/tex] and [tex]\( y = -3 \)[/tex] is:
[tex]\[
2
\][/tex]
