To solve this equation, we can simplify it by first combining the two square roots.
√2^x * √3^x = 216
√(2^x * 3^x) = 216
Now, we can simplify the expression inside the square root by using properties of exponents. Since both 2 and 3 are prime numbers, we can rewrite them as powers of themselves.
2^x 3^x = (2 3)^x
2^x * 3^x = 6^x
Now our equation becomes:
√6^x = 216
Taking the square root of both sides:
6^x = 216^2
6^x = 46656
Next, we can rewrite 46656 as a power of 6:
6^x = 6^6
Since the bases are the same, we can set the exponents equal to each other:
To solve this equation, we can simplify it by first combining the two square roots.
√2^x * √3^x = 216
√(2^x * 3^x) = 216
Now, we can simplify the expression inside the square root by using properties of exponents. Since both 2 and 3 are prime numbers, we can rewrite them as powers of themselves.
2^x 3^x = (2 3)^x
2^x * 3^x = 6^x
Now our equation becomes:
√6^x = 216
Taking the square root of both sides:
6^x = 216^2
6^x = 46656
Next, we can rewrite 46656 as a power of 6:
6^x = 6^6
Since the bases are the same, we can set the exponents equal to each other:
x = 6
Therefore, the solution to the equation is x = 6.