To solve the equation:
[tex]\[
\frac{1}{64 a^3 + 7} - 7 = -\frac{64 a^3}{64 a^3 + 7}
\][/tex]
we need to follow these steps:
1. Clear the fractions by finding a common denominator.
The common denominator of the fractions [tex]\(\frac{1}{64 a^3 + 7}\)[/tex] and [tex]\(\frac{64 a^3}{64 a^3 + 7}\)[/tex] is [tex]\(64 a^3 + 7\)[/tex].
2. Multiply through by the common denominator to get:
[tex]\[
\left( \frac{1}{64 a^3 + 7} - 7 \right) \cdot (64 a^3 + 7) = \left( -\frac{64 a^3}{64 a^3 + 7} \right) \cdot (64 a^3 + 7)
\][/tex]
3. Simplify each term on both sides:
On the left side:
[tex]\[
\left( \frac{1}{64 a^3 + 7} \cdot (64 a^3 + 7) \right) - 7 (64 a^3 + 7) = 1 - 7 (64 a^3 + 7)
\][/tex]
On the right side:
[tex]\[
-64 a^3
\][/tex]
4. Rewrite the equation:
[tex]\[
1 - 7 (64 a^3 + 7) = -64 a^3
\][/tex]
Expanding the left side:
[tex]\[
1 - 448 a^3 - 49 = -64 a^3
\][/tex]
5. Combine like terms:
[tex]\[
1 - 49 - 448 a^3 = -64 a^3
\][/tex]
This simplifies to:
[tex]\[
-48 - 448 a^3 = -64 a^3
\][/tex]
6. Isolate the [tex]\(a^3\)[/tex] terms:
[tex]\[
-448 a^3 + 64 a^3 = -48
\][/tex]
Combine the [tex]\(a^3\)[/tex] terms:
[tex]\[
-384 a^3 = -48
\][/tex]
7. Solve for [tex]\(a^3\)[/tex]:
[tex]\[
a^3 = \frac{-48}{-384} = \frac{1}{8}
\][/tex]
8. Find [tex]\(a\)[/tex].
Taking the cube root of both sides:
[tex]\[
a = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}
\][/tex]
However, solving the equation yields the complex roots as well:
[tex]\[
a = -\frac{1}{2}, \quad a = \frac{1}{4} - \frac{\sqrt{3}i}{4}, \quad a = \frac{1}{4} + \frac{\sqrt{3}i}{4}
\][/tex]
So, the solutions to the equation are:
[tex]\[
a = -\frac{1}{2}, \quad a = \frac{1}{4} - \frac{\sqrt{3}i}{4}, \quad a = \frac{1}{4} + \frac{\sqrt{3}i}{4}
\][/tex]
