Add. $\newline$ The numerator should be expanded and simplified. The denominator should be either expanded or factored. $\newline$ $\frac{7}{x+4}+\frac{3}{x+6}=$

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Grade 8

Negative Exponents

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Q. Add.

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The numerator should be expanded and simplified. The denominator should be either expanded or factored.

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\frac{7}{x+4}+\frac{3}{x+6}=

Identify common denominator: Identify the common denominator for the two fractions. $\newline$ To add the fractions, we need a common denominator. The least common denominator (LCD) for $(x+4)$ and $(x+6)$ is the product of the two, since they are distinct linear factors. $\newline$ Common Denominator: $(x+4)(x+6)$
Rewrite fractions with common denominator: Rewrite each fraction with the common denominator. $\newline$ We need to adjust each fraction so that they both have the common denominator of $(x+4)(x+6)$ . $\newline$ First Fraction: $\frac{7}{x+4} \times \frac{x+6}{x+6} = \frac{7(x+6)}{(x+4)(x+6)}$ $\newline$ Second Fraction: $\frac{3}{x+6} \times \frac{x+4}{x+4} = \frac{3(x+4)}{(x+4)(x+6)}$
Expand numerators of fractions: Expand the numerators of both fractions. $\newline$ Now we expand the numerators to simplify the fractions. $\newline$ First Fraction's Numerator: $7(x+6) = 7x + 42$ $\newline$ Second Fraction's Numerator: $3(x+4) = 3x + 12$
Combine fractions: Combine the fractions. $\newline$ Now that both fractions have the same denominator, we can combine them by adding their numerators. $\newline$ Sum: $(7x + 42 + 3x + 12)/((x+4)(x+6))$
Simplify numerator of combined fraction: Simplify the numerator of the combined fraction. Combine like terms in the numerator. Simplified Numerator: $(7x + 3x) + (42 + 12) = 10x + 54$
Write final simplified fraction: Write the final simplified fraction. $\newline$ The combined fraction with the simplified numerator is: $\newline$ Final Answer: $(10x + 54)/((x+4)(x+6))$