Simplify the expression: $\frac{11 u^6 v^3}{55 u^2 v^8}$
Answer (1)
Simplify the numerical coefficients: 55 11 = 5 1 .
Simplify the powers of u : u 2 u 6 = u 6 − 2 = u 4 .
Simplify the powers of v : v 8 v 3 = v 3 − 8 = v − 5 = v 5 1 .
Combine the simplified terms: 5 1 u 4 ⋅ v 5 1 = 5 v 5 u 4 .
Explanation
Understanding the Problem We are asked to simplify the expression 55 u 2 v 8 11 u 6 v 3 . This involves simplifying the numerical coefficients and applying the rules of exponents to the variables u and v .
Simplifying Coefficients First, we simplify the fraction formed by the coefficients: 55 11 = 11 × 5 11 × 1 = 5 1 So, the expression becomes 5 1 ⋅ u 2 v 8 u 6 v 3 .
Simplifying Powers of u Next, we simplify the powers of u . Using the rule x b x a = x a − b , we have u 2 u 6 = u 6 − 2 = u 4 .
Simplifying Powers of v Now, we simplify the powers of v . Using the same rule, we have v 8 v 3 = v 3 − 8 = v − 5 .
Combining Simplified Terms Combining these simplified terms, we get 5 1 u 4 v − 5 . We can rewrite v − 5 as v 5 1 , so the expression becomes 5 1 u 4 ⋅ v 5 1 = 5 v 5 u 4 .
Final Answer Therefore, the simplified expression is 5 v 5 u 4 .
Examples
In physics, when dealing with quantities that have units raised to powers (like meters per second squared for acceleration), simplifying expressions like this helps in understanding the relationships between different physical quantities. For example, if u represents distance and v represents time, the simplified expression could relate to a derived quantity involving distance and time.
