Simplify: $\frac{3^{3 x+2}-3^{3 x+1}}{6 \times 27^x}$
Answer (1)
Factor out 3 3 x from the numerator: 3 3 x + 2 − 3 3 x + 1 = 3 3 x ( 9 − 3 ) = 6 × 3 3 x .
Rewrite the denominator: 6 × 2 7 x = 6 × ( 3 3 ) x = 6 × 3 3 x .
Simplify the expression: 6 × 3 3 x 6 × 3 3 x .
Cancel out the common factors: The simplified expression is 1 .
Explanation
Analyze the expression We are asked to simplify the expression 6 × 2 7 x 3 3 x + 2 − 3 3 x + 1 . Let's analyze the numerator and denominator separately to identify common factors and simplify the expression.
Rewrite the numerator First, let's rewrite the terms in the numerator using exponent rules. Recall that a m + n = a m × a n . Thus, we can rewrite 3 3 x + 2 as 3 3 x × 3 2 and 3 3 x + 1 as 3 3 x × 3 1 . So the numerator becomes: 3 3 x + 2 − 3 3 x + 1 = 3 3 x × 3 2 − 3 3 x × 3 1 = 3 3 x × 9 − 3 3 x × 3
Factor the numerator Now, we can factor out the common term 3 3 x from the numerator: 3 3 x × 9 − 3 3 x × 3 = 3 3 x ( 9 − 3 ) = 3 3 x ( 6 ) = 6 × 3 3 x
Simplify the denominator Next, let's simplify the denominator. We have 6 × 2 7 x . We can rewrite 27 as 3 3 , so 2 7 x = ( 3 3 ) x = 3 3 x . Therefore, the denominator becomes: 6 × 2 7 x = 6 × 3 3 x
Rewrite the expression Now, we can rewrite the entire expression as: 6 × 2 7 x 3 3 x + 2 − 3 3 x + 1 = 6 × 3 3 x 6 × 3 3 x
Cancel common factors Finally, we can cancel out the common factors in the numerator and the denominator. We have a factor of 6 and a factor of 3 3 x in both the numerator and the denominator. Canceling these factors, we get: 6 × 3 3 x 6 × 3 3 x = 1
Final Answer Therefore, the simplified expression is 1.
Examples
This simplification skill is useful in various fields, such as physics and engineering, where exponential functions are used to model growth and decay processes. For instance, simplifying expressions involving radioactive decay rates or compound interest calculations can make complex problems more manageable. By simplifying the expression, we can easily understand the relationship between variables and make predictions or decisions based on the model.
