Final answer:
To solve the equation 36³=6⁴, we utilize the properties of exponents, specifically the cubing of exponentials rule. By recognizing that 36 is a power of 6, we can simplify the equation to 6⁶=6⁴. This allows us to find that x is equal to 7, by setting the exponents equal to each other and solving for x.
Step-by-step explanation:
To solve the equation 36³=6⁴, we need to recognize the relationship between the numbers 36 and 6 in terms of their prime factorization. Both numbers can be expressed as powers of 6 since 36 is simply 6². This knowledge helps us rewrite the equation using the cubing of exponentials rule which states that when we cube a number, we also multiply its exponent by 3.
The equation 36³=6⁴ can be re-expressed as (6²)³=6⁴. Applying the rule, we can simplify the left side of the equation: (6²)³=6²·³=6⁶. Now we have 6⁶=6⁴, which implies that the exponents must be equal because the base numbers are the same. Therefore, x-1 = 6.
Adding 1 to both sides results in x = 6 + 1. Therefore, x = 7. We have solved the exponential equation through the method of equalizing exponents.