Sure! Let's solve the equation [tex]\(16^{-7 x} = 4^{4 x + 5}\)[/tex] step-by-step.
To solve this equation, we'll first rewrite the bases of the exponential terms using a common base.
1. Rewrite the bases in terms of 2:
- [tex]\(16\)[/tex] can be written as [tex]\(2^4\)[/tex].
- [tex]\(4\)[/tex] can be written as [tex]\(2^2\)[/tex].
Therefore, we can rewrite the original equation as:
[tex]\[
(2^4)^{-7x} = (2^2)^{4x + 5}
\][/tex]
2. Simplify the exponents:
- When raising one power to another, we multiply the exponents.
[tex]\[
2^{4 \cdot (-7x)} = 2^{2 \cdot (4x + 5)}
\][/tex]
Simplifying the exponents on both sides:
[tex]\[
2^{-28x} = 2^{8x + 10}
\][/tex]
3. Set the exponents equal:
Since the bases are the same (both are base 2), we can set the exponents equal to each other:
[tex]\[
-28x = 8x + 10
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- First, get all the terms involving [tex]\(x\)[/tex] on one side of the equation:
[tex]\[
-28x - 8x = 10
\][/tex]
- Combine like terms:
[tex]\[
-36x = 10
\][/tex]
- Solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(-36\)[/tex]:
[tex]\[
x = \frac{10}{-36}
\][/tex]
- Simplify the fraction:
[tex]\[
x = -\frac{5}{18}
\][/tex]
So, the solution for the equation [tex]\(16^{-7 x} = 4^{4 x + 5}\)[/tex] is:
[tex]\[
x = -\frac{5}{18}
\][/tex]
