To find the value of [tex]\( k \)[/tex] in the equation [tex]\( 81y^2 - k = (9y + \frac{1}{2})(9y - \frac{1}{2}) \)[/tex], we can begin by simplifying the right-hand side of the equation.
1. Simplify the Right-Hand Side:
We have to expand the expression [tex]\( (9y + \frac{1}{2})(9y - \frac{1}{2}) \)[/tex].
Recall the difference of squares formula:
[tex]\[
(a + b)(a - b) = a^2 - b^2
\][/tex]
Here, [tex]\( a = 9y \)[/tex] and [tex]\( b = \frac{1}{2} \)[/tex].
Applying the difference of squares formula:
[tex]\[
(9y + \frac{1}{2})(9y - \frac{1}{2}) = (9y)^2 - \left(\frac{1}{2}\right)^2
\][/tex]
2. Calculate Each Term:
[tex]\[
(9y)^2 = 81y^2
\][/tex]
[tex]\[
\left(\frac{1}{2}\right)^2 = \frac{1}{4}
\][/tex]
Substitution these values in the equation:
[tex]\[
(9y + \frac{1}{2})(9y - \frac{1}{2}) = 81y^2 - \frac{1}{4}
\][/tex]
3. Setting the Expanded Form Equal to the Left-Hand Side:
The given equation is:
[tex]\[
81y^2 - k = 81y^2 - \frac{1}{4}
\][/tex]
Since [tex]\( 81y^2 \)[/tex] is on both sides, we can cancel them out, leaving:
[tex]\[
-k = -\frac{1}{4}
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{1}{4}
\][/tex]
Thus, the value of [tex]\( k \)[/tex] is [tex]\(\boxed{\frac{1}{4}}\)[/tex].
