Perform the indicated operation.
$\frac{3 x^5}{7}-\frac{5 x^3}{21}= \square$ (Simplify your answer.)
Answer (1)
Find a common denominator: Rewrite the fractions with a common denominator of 21.
Subtract the fractions: Combine the fractions over the common denominator.
Factor the numerator: Factor out the greatest common factor from the numerator.
Simplify the expression: The simplified expression is 21 x 3 ( 9 x 2 − 5 ) .
Explanation
Understanding the Problem We are asked to subtract two terms: 7 3 x 5 and 21 5 x 3 . To do this, we need to find a common denominator.
Finding a Common Denominator The least common denominator (LCD) of 7 and 21 is 21. We rewrite the first fraction with the LCD: 7 3 x 5 = 7 × 3 3 x 5 × 3 = 21 9 x 5 .
Subtracting the Fractions Now we can subtract the two fractions: 21 9 x 5 − 21 5 x 3 = 21 9 x 5 − 5 x 3 .
Factoring the Numerator We can factor out the greatest common factor (GCF) from the numerator, which is x 3 : 21 9 x 5 − 5 x 3 = 21 x 3 ( 9 x 2 − 5 ) .
Final Answer The expression is now simplified. Therefore, the final answer is: 21 x 3 ( 9 x 2 − 5 ) .
Examples
Understanding how to combine and simplify expressions like this is fundamental in many areas, such as physics and engineering, where you might need to combine different forces or signals. For example, if you are analyzing the motion of an object affected by two different forces that vary with position, you might end up with an expression similar to the one in this problem. Simplifying such expressions allows you to more easily understand and predict the object's behavior. This skill is also crucial in computer graphics, where combining transformations requires simplifying polynomial expressions to optimize rendering performance.
