Evaluate (-m^5n^2-1/2m^4n^4+2/3m^3n-4mn^4)÷(-4m^5n^3)
The question is asking for the simplification of a given algebraic expression that involves variables with exponents and coefficients, including fractions. The operation to be performed is division of two polynomial expressions. Specifically, you are to divide the polynomial (-m^5n^2 - 1/2m^4n^4 + 2/3m^3n - 4mn^4) by the monomial (-4m^5n^3). The answer should be obtained by applying the laws of exponents and the rules of dividing polynomials, particularly focusing on how to divide each term of the polynomial by the monomial.
$\left(\right. - m^{5} n^{2} - \frac{1}{2} m^{4} n^{4} + \frac{2}{3} m^{3} n - 4 m n^{4} \left.\right) \div \left(\right. - 4 m^{5} n^{3} \left.\right)$
Answer
Solution:
Step 1: Convert Division to Fraction
Convert the given division expression into a fraction:
$$\frac{-m^5n^2 - \frac{1}{2}m^4n^4 + \frac{2}{3}m^3n - 4mn^4}{-4m^5n^3}$$
Step 2: Simplify the Numerator
Step 2.1: Factor Out Common Terms
Factor out the common term $mn$ from each term in the numerator:
$$\frac{mn(-m^4n - \frac{1}{2}m^3n^3 + \frac{2}{3}m^2 - 4n^3)}{-4m^5n^3}$$
Step 2.2: Combine Like Terms
Combine like terms by finding common denominators and simplifying:
$$\frac{mn\left(\frac{-2m^4n - m^3n^3 + \frac{4}{3}m^2 - 24n^3}{6}\right)}{-4m^5n^3}$$
Step 3: Multiply by Reciprocal
Multiply the simplified numerator by the reciprocal of the denominator:
$$\frac{mn\left(\frac{-2m^4n - m^3n^3 + \frac{4}{3}m^2 - 24n^3}{6}\right)}{1} \times \frac{1}{-4m^5n^3}$$
Step 4: Cancel Common Factors
Cancel out common factors in the numerator and denominator:
$$\frac{\left(\frac{-2m^4n - m^3n^3 + \frac{4}{3}m^2 - 24n^3}{6}\right)}{-4m^4n^2}$$
Step 5: Simplify the Fraction
Simplify the fraction by combining the terms and reducing:
$$\frac{6m^4n + 3m^3n^3 - 4m^2 + 24n^3}{24m^4n^2}$$
Knowledge Notes:
Algebraic Manipulation: The process involves manipulating algebraic expressions by factoring, expanding, and simplifying to reach the desired form.
Factoring: Extracting common factors from terms to simplify expressions. For example, factoring $mn$ from $-m^5n^2$ yields $mn(-m^4n)$.
Combining Like Terms: Terms with the same variables and exponents can be combined by adding or subtracting their coefficients.
Common Denominators: To combine fractions, we find a common denominator, which is usually the product of the denominators of the fractions being combined.
Reciprocal: The reciprocal of a number (or algebraic expression) is 1 divided by that number (or expression). Multiplying by the reciprocal is equivalent to division.
Cancellation: Common factors in the numerator and denominator of a fraction can be cancelled out to simplify the fraction.
Negative Signs: Negative signs can be distributed across terms in parentheses or factored out, depending on the expression's structure.
Power Rule: When multiplying variables with the same base, add the exponents ($a^m \cdot a^n = a^{m+n}$).
Distributive Property: Multiplying a term outside the parentheses by each term inside the parentheses ($a(b + c) = ab + ac$).
Commutative Property: The order of addition or multiplication does not affect the result ($a + b = b + a$ and $ab = ba$).
