To solve the given equation
[tex]\[
\frac{5^{3x} \times 25}{5^x} = 5^3 \times 125
\][/tex]
we will follow a detailed, step-by-step approach.
### Step 1: Simplify the Left-Hand Side (LHS) of the Equation
The left-hand side of the equation is
[tex]\[
\frac{5^{3x} \times 25}{5^x}
\][/tex]
First, note that [tex]\(25\)[/tex] can be written as [tex]\(5^2\)[/tex]. Therefore, we get:
[tex]\[
\frac{5^{3x} \times 5^2}{5^x}
\][/tex]
Next, we use the property of exponents that allows us to combine powers:
[tex]\[
a^m \times a^n = a^{m+n}
\][/tex]
Applying this to the numerator, we get:
[tex]\[
5^{3x} \times 5^2 = 5^{3x + 2}
\][/tex]
Thus, the left-hand side now becomes:
[tex]\[
\frac{5^{3x+2}}{5^x}
\][/tex]
We can simplify this further using the property of exponents that allows us to divide powers:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
Applying this property:
[tex]\[
\frac{5^{3x+2}}{5^x} = 5^{(3x+2) - x} = 5^{2x + 2}
\][/tex]
So, the simplified left-hand side is:
[tex]\[
5^{2x + 2}
\][/tex]
### Step 2: Simplify the Right-Hand Side (RHS) of the Equation
The right-hand side of the equation is:
[tex]\[
5^3 \times 125
\][/tex]
Note that [tex]\(125\)[/tex] can be written as [tex]\(5^3\)[/tex]. Therefore, we get:
[tex]\[
5^3 \times 5^3
\][/tex]
Using the property of exponents mentioned earlier:
[tex]\[
a^m \times a^n = a^{m+n}
\][/tex]
Applying this:
[tex]\[
5^3 \times 5^3 = 5^{3+3} = 5^6
\][/tex]
So, the simplified right-hand side is:
[tex]\[
5^6
\][/tex]
### Step 3: Equate the Simplified LHS and RHS
Equating the simplified left-hand side and right-hand side, we get:
[tex]\[
5^{2x + 2} = 5^6
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
2x + 2 = 6
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
To solve for [tex]\(x\)[/tex]:
[tex]\[
2x + 2 = 6
\][/tex]
Subtract 2 from both sides:
[tex]\[
2x = 6 - 2
\][/tex]
Simplifying, we get:
[tex]\[
2x = 4
\][/tex]
Divide both sides by 2:
[tex]\[
x = \frac{4}{2} = 2
\][/tex]
Thus, the solution is:
[tex]\[
x = 2
\][/tex]
