Perform the division.
$\frac{-30 x^6+100 x^7}{25 x^7}= \square$ (Simplify your answer.)
Answer (2)
The division of 25 x 7 − 30 x 6 + 100 x 7 simplifies to 5 x 20 x − 6 . The simplification involves dividing each term by the denominator and combining the results into a single fraction. The final answer is 5 x 20 x − 6 .
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Divide each term in the numerator by the denominator: 25 x 7 − 30 x 6 + 25 x 7 100 x 7 .
Simplify each term: − 5 x 6 + 4 .
Combine the terms: 5 x − 6 + 20 x .
The simplified expression is 5 x 20 x − 6 .
Explanation
Understanding the problem We are asked to perform the division and simplify the expression 25 x 7 − 30 x 6 + 100 x 7 .
Dividing each term We can divide each term in the numerator by the denominator: 25 x 7 − 30 x 6 + 100 x 7 = 25 x 7 − 30 x 6 + 25 x 7 100 x 7
Simplifying each term Now, we simplify each term separately. For the first term, we have:$ 25 x 7 − 30 x 6 = − 25 30 ⋅ x 7 x 6 = − 5 6 ⋅ x 6 − 7 = − 5 6 x − 1 = − 5 x 6 F or t h eseco n d t er m , w e ha v e : $ \frac{100 x^7}{25 x^7} = \frac{100}{25} \cdot \frac{x^7}{x^7} = 4 \cdot x^{7-7} = 4 \cdot x^0 = 4 \cdot 1 = 4
Combining the terms Combining the simplified terms, we get:$ 25 x 7 − 30 x 6 + 100 x 7 = − 5 x 6 + 4 $We can write this as a single fraction by finding a common denominator, which is 5 x :$ − 5 x 6 + 4 = − 5 x 6 + 5 x 4 ( 5 x ) = 5 x − 6 + 20 x = 5 x 20 x − 6 $
Final Answer Thus, the simplified expression is 5 x 20 x − 6 .
Examples
Understanding how to simplify rational expressions like this is useful in many areas, such as physics and engineering, where complex formulas often need to be simplified to make calculations easier. For example, when analyzing the motion of an object, you might encounter a complex expression involving variables like time and distance. Simplifying this expression can help you understand the relationship between these variables more clearly and make predictions about the object's future motion. It's also useful in computer graphics when dealing with transformations and scaling of objects.
