To solve the equation [tex]\( 125^{(x+1)} - 5^{3x} - 5 = 615 \)[/tex], let’s break it down into manageable steps.
### Step 1: Identify Properties
Notice that [tex]\( 125 \)[/tex] can be written as [tex]\( 5^3 \)[/tex]. Hence, we can rewrite [tex]\( 125^{(x+1)} \)[/tex] as:
[tex]\[ 125^{(x+1)} = (5^3)^{(x+1)} = 5^{3(x+1)} \][/tex]
### Step 2: Substitute and Simplify
Now, substitute [tex]\( 5^{3(x+1)} \)[/tex] back into the original equation:
[tex]\[ 5^{3(x+1)} - 5^{3x} - 5 = 615 \][/tex]
Next, expand [tex]\( 5^{3(x+1)} \)[/tex]:
[tex]\[ 5^{3(x+1)} = 5^{3x + 3} = 5^{3x} \cdot 5^3 = 5^{3x} \cdot 125 \][/tex]
Thus, the equation becomes:
[tex]\[ 125 \cdot 5^{3x} - 5^{3x} - 5 = 615 \][/tex]
### Step 3: Factor Common Terms
Factor [tex]\( 5^{3x} \)[/tex] out of the terms:
[tex]\[ 5^{3x} \cdot 125 - 5^{3x} - 5 = 615 \][/tex]
[tex]\[ 5^{3x} (125 - 1) - 5 = 615 \][/tex]
[tex]\[ 5^{3x} \cdot 124 - 5 = 615 \][/tex]
Add 5 to both sides to simplify further:
[tex]\[ 124 \cdot 5^{3x} = 620 \][/tex]
### Step 4: Isolate the Exponential Term
Divide both sides by 124:
[tex]\[ 5^{3x} = \frac{620}{124} \][/tex]
[tex]\[ 5^{3x} = 5 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Since the bases are the same [tex]\( (5) \)[/tex], set the exponents equal to each other:
[tex]\[ 3x = 1 \][/tex]
[tex]\[ x = \frac{1}{3} \][/tex]
Thus, the solution to the equation [tex]\( 125^{(x+1)} - 5^{3x} - 5 = 615 \)[/tex] is:
[tex]\[ x = \frac{1}{3} \][/tex]
Hence, [tex]\( x = \frac{1}{3} \)[/tex] is the value that satisfies the given equation.
