Certainly! I'll guide you through solving the equation step-by-step.
The equation given is:
[tex]\[ 4^{x+1} = 64^{x-5} \][/tex]
### Step 1: Simplify 64 as a power of 4
First, we recognize that 64 can be expressed as a power of 4. Since [tex]\(64 = 4^3\)[/tex], we rewrite the equation:
[tex]\[ 4^{x+1} = (4^3)^{x-5} \][/tex]
### Step 2: Use the properties of exponents
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify the right-hand side of the equation:
[tex]\[ (4^3)^{x-5} = 4^{3(x-5)} \][/tex]
Substitute this back into the equation:
[tex]\[ 4^{x+1} = 4^{3(x-5)} \][/tex]
### Step 3: Equate the exponents
Since the bases are the same (both are 4), we can set the exponents equal to each other:
[tex]\[ x + 1 = 3(x - 5) \][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
Now, we solve the equation for [tex]\(x\)[/tex]:
First, distribute the 3 on the right-hand side:
[tex]\[ x + 1 = 3x - 15 \][/tex]
Next, isolate [tex]\(x\)[/tex] by moving the terms involving [tex]\(x\)[/tex] to one side and the constant terms to the other side. Subtract [tex]\(x\)[/tex] from both sides of the equation:
[tex]\[ 1 = 2x - 15 \][/tex]
Add 15 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ 16 = 2x \][/tex]
Finally, divide by 2:
[tex]\[ x = 8 \][/tex]
Therefore, the solution to the equation [tex]\( 4^{x+1} = 64^{x-5} \)[/tex] is:
[tex]\[ x = 8 \][/tex]
