Simplify (30x^7y^4 + 9x^5y^5 - 54x^8y^3) / (6x^3y^2)
Understand the Problem
The question asks to simplify an algebraic expression. The numerator consists of three terms, and the denominator is a single term. We need to divide each term in the numerator by the denominator and simplify the exponents.
Answer
$5x^4y^2 + \frac{3}{2}x^2y^3 - 9x^5y$
Answer for screen readers
$5x^4y^2 + \frac{3}{2}x^2y^3 - 9x^5y$
Steps to Solve
- Rewrite the expression
Rewrite the expression as a sum of three fractions: $$ \frac{30x^7y^4 + 9x^5y^5 - 54x^8y^3}{6x^3y^2} = \frac{30x^7y^4}{6x^3y^2} + \frac{9x^5y^5}{6x^3y^2} - \frac{54x^8y^3}{6x^3y^2} $$
- Simplify each fraction
Simplify each fraction by dividing the coefficients and using the quotient rule for exponents ($ \frac{x^a}{x^b} = x^{a-b} $):
$$ \frac{30x^7y^4}{6x^3y^2} = 5x^{7-3}y^{4-2} = 5x^4y^2 $$
$$ \frac{9x^5y^5}{6x^3y^2} = \frac{3}{2}x^{5-3}y^{5-2} = \frac{3}{2}x^2y^3 $$
$$ \frac{54x^8y^3}{6x^3y^2} = 9x^{8-3}y^{3-2} = 9x^5y $$
- Combine the simplified terms
Combine the simplified terms to get the final expression: $$ 5x^4y^2 + \frac{3}{2}x^2y^3 - 9x^5y $$
$5x^4y^2 + \frac{3}{2}x^2y^3 - 9x^5y$
More Information
The simplified expression is a polynomial in two variables, $x$ and $y$.
Tips
A common mistake is to incorrectly apply the quotient rule for exponents, either by subtracting the exponents in the wrong order or by making arithmetic errors. Also, forgetting to simplify the coefficients of each term is a common mistake.
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