Perform the following operation and express in simplest form. (x)/(x^(2)+2x-48)÷(x^(3))/(x^(2)-64) Answer:

Factor Denominators: First, let's factor the denominators of both fractions to simplify the expression. The denominator of the first fraction is x^2 + 2x - 48, which factors into (x + 8)(x - 6). The denominator of the second fraction is x^2 - 64, which is a difference of squares and factors into (x + 8)(x - 8).Rewrite with Factored Denominators: Now, let's rewrite the original expression with the factored denominators. (x)/(x^2 + 2x - 48) ÷ (x^3)/(x^2 - 64) becomes (x)/((x + 8)(x - 6)) ÷ (x^3)/((x + 8)(x - 8))Convert to Multiplication: Next, we will convert the division of fractions into multiplication by the reciprocal of the second fraction. (x)/((x + 8)(x - 6)) * ((x + 8)(x - 8))/(x^3)Cancel Common Factors: Now, we can cancel out the common factors in the numerator and the denominator. The (x + 8) term is present in both the numerator and the denominator, so we can cancel it out. (x)/((x - 6)) * (1)/(x^2)Simplify Further: We can now simplify the expression further by canceling out the x term in the numerator of the first fraction and the x^2 term in the denominator of the second fraction. This leaves us with: 1/((x - 6)) * (1)/(x)Multiply Remaining Fractions: Finally, we multiply the remaining fractions to get the simplified expression. 1/((x - 6)x)

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Perform the following operation and express in simplest form.

(x)/(x^(2)+2x-48)÷(x^(3))/(x^(2)-64)
Answer:

Perform the following operation and express in simplest form. $\newline$ $\frac{x}{x^{2}+2 x-48} \div \frac{x^{3}}{x^{2}-64}$ $\newline$ Answer:

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Math Problems

Algebra 1

Multiplication with rational exponents

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Q. Perform the following operation and express in simplest form.

\newline

\frac{x}{x^{2}+2 x-48} \div \frac{x^{3}}{x^{2}-64}

\newline

Answer:

Factor Denominators: First, let's factor the denominators of both fractions to simplify the expression. $\newline$ The denominator of the first fraction is $x^2 + 2x - 48$ , which factors into $(x + 8)(x - 6)$ . $\newline$ The denominator of the second fraction is $x^2 - 64$ , which is a difference of squares and factors into $(x + 8)(x - 8)$ .
Rewrite with Factored Denominators: Now, let's rewrite the original expression with the factored denominators. $\newline$ $\frac{x}{x^2 + 2x - 48}$ ÷ $\frac{x^3}{x^2 - 64}$ $\newline$ becomes $\newline$ $\frac{x}{(x + 8)(x - 6)}$ ÷ $\frac{x^3}{(x + 8)(x - 8)}$
Convert to Multiplication: Next, we will convert the division of fractions into multiplication by the reciprocal of the second fraction. $\frac{x}{(x + 8)(x - 6)} \times \frac{(x + 8)(x - 8)}{x^3}$
Cancel Common Factors: Now, we can cancel out the common factors in the numerator and the denominator. $\newline$ The $(x + 8)$ term is present in both the numerator and the denominator, so we can cancel it out. $\newline$ $\frac{x}{(x - 6)} \times \frac{1}{x^2}$
Simplify Further: We can now simplify the expression further by canceling out the $x$ term in the numerator of the first fraction and the $x^2$ term in the denominator of the second fraction. $\newline$ This leaves us with: $\newline$ $\frac{1}{(x - 6)} \times \frac{1}{x}$
Multiply Remaining Fractions: Finally, we multiply the remaining fractions to get the simplified expression. $\newline$ $\frac{1}{((x - 6)x)}$