2^x × 3^x = 5^(x + 1)
To solve the given equation, we need to rewrite the equation using the same base. We can rewrite "5" as "2^2.32" since it is between 2^2 and 2^3.
Therefore, the equation becomes:
2^x × 3^x = (2^2.32)^(x + 1)
Now, apply the power rule by multiplying the exponents:
2^x × 3^x = 2^(2.32(x + 1))
Since the bases are the same, we can equate the exponents:
x = 2.32(x + 1)
Distribute 2.32:
x = 2.32x + 2.32
Subtract 2.32x from both sides:
x - 2.32x = 2.32
Combine like terms:
-1.32x = 2.32
Divide both sides by -1.32:
x = 2.32 / -1.32
Simplifying the division gives us:
x ≈ -1.758
Therefore, the solution to the equation 2^x × 3^x = 5^(x + 1) is x ≈ -1.758.
To solve the equation 2^x × 3^x = 5^(x + 1), we need to simplify both sides of the equation and then isolate the variable x.
Step 1: Rewrite the expressions using the rules of exponents.
2^x × 3^x = 5^(x + 1)
Rewriting the exponents in terms of multiplication, we have:
(2 × 3)^x = 5 × 5^x
Step 2: Simplify both sides of the equation.
Multiplying 2 and 3, we get:
6^x = 5 × 5^x
Step 3: Rewrite the equation using the same base.
Since both sides have the same base (5), we can rewrite the equation as:
6^x = 5^(1 + x)
Step 4: Set the exponents equal to each other.
Since the bases are equal, we can equate the exponents:
x = 1 + x
Step 5: Solve for x.
To isolate the variable x, we can subtract x from both sides of the equation:
x - x = 1 + x - x
0 = 1
Step 6: Analyze the result.
Since the equation 0 = 1 is not true, there is no solution to the equation 2^x × 3^x = 5^(x + 1). Therefore, the original equation has no solution.