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(3/4)^6 X (16/9)^5 = (4/3)^x+2

2 min read Jun 16, 2024
(3/4)^6 X (16/9)^5 = (4/3)^x+2

Solving the Equation (3/4)^6 x (16/9)^5 = (4/3)^x+2

This problem involves simplifying expressions with fractional exponents and solving for an unknown exponent. Here's how we can break it down:

Step 1: Simplifying the Exponents

  • Rewrite (16/9) as (4/3)²: This allows us to have a common base with the right side of the equation.
  • Rewrite (3/4)^6 as (4/3)^-6: This is because (a/b)^-n = (b/a)^n.

Now the equation becomes:

(4/3)^-6 x ((4/3)²)⁵ = (4/3)^x+2

Step 2: Applying Exponent Rules

  • Multiply exponents with the same base: (a^m) * (a^n) = a^(m+n)

The left side of the equation simplifies to:

(4/3)^(-6+10) = (4/3)^4

Step 3: Solving for x

  • Equate exponents: Since the bases are the same, we can equate the exponents.

(4/3)^4 = (4/3)^x+2

This gives us: 4 = x + 2

Step 4: Finding the Value of x

  • Solve for x: Subtract 2 from both sides of the equation.

x = 2

Therefore, the solution to the equation (3/4)^6 x (16/9)^5 = (4/3)^x+2 is x = 2.

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