To solve the equation [tex]\( f(x) = 0 \)[/tex], where [tex]\( f(x) = 16x^4 - 81 \)[/tex], we will follow these steps:
1. Start with the given equation:
[tex]\[
16x^4 - 81 = 0
\][/tex]
2. Add 81 to both sides of the equation to isolate the quartic term:
[tex]\[
16x^4 = 81
\][/tex]
3. Divide both sides by 16 to solve for [tex]\( x^4 \)[/tex]:
[tex]\[
x^4 = \frac{81}{16}
\][/tex]
4. Take the fourth root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt[4]{\frac{81}{16}}
\][/tex]
5. Simplify the expression under the fourth root:
[tex]\[
x = \pm \sqrt[4]{\left(\frac{9}{4}\right)^2}
\][/tex]
6. Since [tex]\(\sqrt[4]{a^2} = \sqrt{a}\)[/tex], we get:
[tex]\[
x = \pm \sqrt{\frac{9}{4}} = \pm \frac{3}{2}
\][/tex]
Thus, we have two real solutions:
[tex]\[
x = \frac{3}{2} \quad \text{and} \quad x = -\frac{3}{2}
\][/tex]
7. Consider the complex roots by recognizing that taking the fourth root in the complex plane also gives complex solutions:
[tex]\[
x = \pm \frac{3i}{2}
\][/tex]
So, the complete solution set for the equation is:
[tex]\[
x = \frac{3}{2}, \quad x = -\frac{3}{2}, \quad x = \frac{3i}{2}, \quad x = -\frac{3i}{2}
\][/tex]
Hence, the final solutions to the equation [tex]\( 16x^4 - 81 = 0 \)[/tex] are:
[tex]\[
-\frac{3}{2}, \frac{3}{2}, -\frac{3i}{2}, \frac{3i}{2}
\][/tex]
