Sure! Let's solve the given expression step-by-step.
We need to simplify the left-hand side and the right-hand side of the equation:
[tex]\[ \frac{5^{3x} \times 25}{5^x} = 5^3 \times 125 \][/tex]
### Step 1: Simplify the Left-Hand Side
1. Rewrite [tex]\(25\)[/tex] and simplify:
Since [tex]\(25\)[/tex] can be written as [tex]\(5^2\)[/tex], the expression now becomes:
[tex]\[ \frac{5^{3x} \times 5^2}{5^x} \][/tex]
2. Apply exponents rule:
Using the exponent rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex], we combine the exponents in the numerator:
[tex]\[ \frac{5^{3x + 2}}{5^x} \][/tex]
3. Simplify the fraction:
Using the exponent rule [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[ 5^{(3x + 2) - x} = 5^{2x + 2} \][/tex]
### Step 2: Simplify the Right-Hand Side
1. Rewrite [tex]\(125\)[/tex]:
Knowing [tex]\(125\)[/tex] is [tex]\(5^3\)[/tex], we write:
[tex]\[ 5^3 \times 125 = 5^3 \times 5^3 \][/tex]
2. Combine exponents:
Again, using the exponent rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
[tex]\[ 5^3 \times 5^3 = 5^{3 + 3} = 5^6 \][/tex]
### Step 3: Comparing Both Sides
Comparing the simplified forms of both sides,
- Left side: [tex]\( 5^{2x + 2} \)[/tex]
- Right side: [tex]\( 5^6 \)[/tex]
If these sides are equal, their exponents must be equal:
[tex]\[ 2x + 2 = 6 \][/tex]
### Solve for [tex]\(x\)[/tex]:
Solve the equation [tex]\(2x + 2 = 6\)[/tex]:
[tex]\[ 2x + 2 = 6 \][/tex]
[tex]\[ 2x = 6 - 2 \][/tex]
[tex]\[ 2x = 4 \][/tex]
[tex]\[ x = 2 \][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the given expression is [tex]\( x = 2 \)[/tex].
To validate this result numerically, the initial equation can be evaluated as follows:
[tex]\[ \left(\frac{5^{3 \cdot 2} \times 25}{5^2} = 5^3 \times 125\right) \][/tex]
[tex]\[ \left(\frac{5^6 \times 25}{5^2} = 5^3 \times 125\right) \][/tex]
[tex]\[ \left(\frac{5^6 \times 5^2}{5^2} = 5^3 \times 5^3\right) \][/tex]
[tex]\[ 5^6 = 5^6 \][/tex]
This confirms the correctness of the derived value [tex]\( x = 2 \)[/tex].
