Question

In Exercises 33–68, add or subtract as indicated. ﻿x+910x3+1115x2\frac{x + 9}{$$10x^3$$} + \frac{11}{$$15x^2$$}
10x3x+9​+15x211​

Accepted Answer

Identify the two rational expressions to be added: $$\frac{x+9}{10x$$^{3}$$}$$ and $$\frac{11}{15x$$^{2}$$}. $$Find the least common denominator (LCD) of the two fractions. The denominators are $$10x^{3}$$ and $$15x^{2}. $$Factor each denominator into prime factors: $$10x^{3} = 2$$ 	imes 5 	imes $$x^{3}$$ and $$15x^{2} = 3$$ 	imes 5 	imes $$x^{2}. $$Determine the LCD by taking the highest powers of each prime factor present: for numbers, take $$2$$, $$3$$, and $$5$$; for variables, take $$x^{3}. So$$, the LCD is $$30x^{3}. $$Rewrite each fraction with the LCD as the denominator. For $$\frac{x+9}{10x$$^{3}$$}$$, multiply numerator and denominator by $$3 to $$get denominator $$30x^{3}. $$For $$\frac{11}{15x$$^{2}$$}$$, multiply numerator and denominator by $$2x to $$get denominator $$30x^{3}. $$Add the numerators over the common denominator: $$\frac{3(x+9) + 11(2x)}{30x$$^{3}$$}. $$Then simplify the numerator by distributing and combining like terms.

In Exercises 33–68, add or subtract as indicated. $\frac{x + 9}{10x^3}\) + \(\frac{11}{15x^2}$

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Key Concepts

Adding and Subtracting Rational Expressions

Finding the Least Common Denominator (LCD)

Simplifying Algebraic Expressions

In Exercises 33–68, add or subtract as indicated. x+910x3+1115x2\(\frac{x + 9}{10x^3}\) + \(\frac{11}{15x^2}\)10x3x+9+15x211