To solve the equation [tex]\( f(x) = 16x^4 - 81 = 0 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that satisfy this equation. Let's break it down into detailed steps:
1. Rewrite the Equation: Start with the given equation:
[tex]\[
16x^4 - 81 = 0
\][/tex]
2. Move the Constant Term: Add 81 to both sides of the equation to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
16x^4 = 81
\][/tex]
3. Solve for [tex]\( x^4 \)[/tex]: Divide both sides by 16 to solve for [tex]\( x^4 \)[/tex]:
[tex]\[
x^4 = \frac{81}{16}
\][/tex]
4. Simplify the Fraction: Notice that [tex]\( \frac{81}{16} \)[/tex] is a perfect fourth power, which makes it easier to take the fourth root:
[tex]\[
x^4 = \left(\frac{3}{2}\right)^4
\][/tex]
5. Take the Fourth Root: To find [tex]\( x \)[/tex], take the fourth root of both sides. Remember that taking the fourth root of a number can yield both real and complex solutions:
[tex]\[
x = \pm \frac{3}{2}, \quad x = \pm \frac{3i}{2}
\][/tex]
Thus, the solutions to the equation [tex]\( 16x^4 - 81 = 0 \)[/tex] are:
[tex]\[
x = -\frac{3}{2}, \quad x = \frac{3}{2}, \quad x = -\frac{3i}{2}, \quad x = \frac{3i}{2}
\][/tex]
Therefore, the equivalent solutions for the equation [tex]\( f(x) = 16x^4 - 81 = 0 \)[/tex] are:
[tex]\[
x = -\frac{3}{2}, \quad x = \frac{3}{2}, \quad x = -\frac{3i}{2}, \quad x = \frac{3i}{2}
\][/tex]
These solutions satisfy the original equation.
