(ii) ((125)/(8))times((125)/(8))^(x)=((5)/(2))^(18)

Jasonbradley

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Jun 16, 2024

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Solving the Exponential Equation: (125/8) * (125/8)^x = (5/2)^18

This article will guide you through the steps to solve the exponential equation: (125/8) * (125/8)^x = (5/2)^18.

Step 1: Express all terms with the same base

First, we need to express all terms in the equation with the same base. Notice that:

• (125/8) can be rewritten as (5^3)/(2^3)
• (5/2) is already in its simplest form.

Therefore, the equation becomes:

(5^3)/(2^3) * ((5^3)/(2^3))^x = (5/2)^18

Step 2: Simplify the equation using exponent rules

Applying the exponent rule (a^m)^n = a^(m*n), we simplify the left side of the equation:

(5^3)/(2^3) * (5^(3x))/(2^(3x)) = (5/2)^18

Combining the terms with the same base:

(5^(3+3x))/(2^(3+3x)) = (5/2)^18

Step 3: Equate exponents

Now that both sides of the equation have the same base, we can equate the exponents:

(3 + 3x) = 18

Step 4: Solve for x

Solving the linear equation for x:

• 3x = 15
• x = 5

Solution

Therefore, the solution to the equation (125/8) * (125/8)^x = (5/2)^18 is x = 5.