To solve the equation (√6)^(8x) = 216x³, we can follow these steps:
- Understand the components: The left side of the equation is an exponential expression with a base of the square root of 6, raised to the power of 8x. The right side is a polynomial expression multiplied by 216 in a cubic form.
- Rewrite the expressions if possible: Notice that √6 can be expressed as 6^(1/2). Thus, we can rewrite the left side:
- (√6)^(8x) = (6^(1/2))^(8x) = 6^(4x)
- Therefore, our equation now looks like:
- 6^(4x) = 216x³
- Express 216 as a power of 6: We know that 216 = 6^3 (since 6 x 6 x 6 = 216). Thus:
- 6^(4x) = (6^3)x³
- The equation now is:
- 6^(4x) = 6^3 * x³
- Set the exponents equal to each other: Since both sides have the same base (6), we can set the exponents equal:
- 4x = 3 + log₆(x³)
- Solving for x: To simplify further, use logarithmic properties:
- log₆(x³) = 3 * log₆(x)
- Now, replacing in our equation gives us:
- 4x = 3 + 3 * log₆(x)
- This will usually require numerical methods or graphing to solve for x unless x takes specific values that simplify the logarithm.
- Finding approximate solutions: A common technique is to use numerical methods or graphing tools. Set:
- f(x) = 4x – 3 – 3 * log₆(x)
- By finding where f(x) = 0, you can numerically evaluate possible solutions.
In conclusion, while we transformed the equation and set the exponents equal, often solving for x may require numerical methods or graphing as the next steps. However, through careful transformation and understanding the properties of logarithms and exponents, we’ve laid a clear pathway to reach a solution.