Math Problem Statement

Add/Subtract Rationals: Simplify each expression.1) 2m25m - m - 2n25m 2) 3x - 3 y24x + x + 3 y24x3) 6x - 3 y4x2y + x + y4x2y 4) m + n30m4 - m + 2n30m45) x + 2 y15x - x - 6 y15x 6) 26 + 34b7) 4y4 - 3x4y3 8) 5x4 - 65xy9) 6a2 + 2b3ab 10) 42 y2 + 26 y11) m - 24m3 + 20m2 - m + 64m3 + 20m2 12) k - 418k - 36 - k - 518k - 36

Solution

Let's simplify each given rational expression step-by-step.

1) 2m25m−m−2n25m\frac{2m^2}{5m} - m - \frac{2n^2}{5m}

  1. Rewrite each term with a common denominator of 5m5m: 2m25m−5m25m−2n25m\frac{2m^2}{5m} - \frac{5m^2}{5m} - \frac{2n^2}{5m}
  2. Combine numerators: 2m2−5m2−2n25m=−3m2−2n25m\frac{2m^2 - 5m^2 - 2n^2}{5m} = \frac{-3m^2 - 2n^2}{5m}

2) 3x−3y24x+x+3y24x\frac{3x - 3y^2}{4x} + \frac{x + 3y^2}{4x}

  1. Combine fractions (denominator 4x4x): (3x−3y2)+(x+3y2)4x\frac{(3x - 3y^2) + (x + 3y^2)}{4x}
  2. Simplify numerator: 3x+x−3y2+3y24x=4x4x=1\frac{3x + x - 3y^2 + 3y^2}{4x} = \frac{4x}{4x} = 1

3) 6x−3y4x2y+x+y4x2y\frac{6x - 3y}{4x^2y} + \frac{x + y}{4x^2y}

  1. Combine fractions (denominator 4x2y4x^2y): (6x−3y)+(x+y)4x2y\frac{(6x - 3y) + (x + y)}{4x^2y}
  2. Simplify numerator: 6x+x−3y+y4x2y=7x−2y4x2y\frac{6x + x - 3y + y}{4x^2y} = \frac{7x - 2y}{4x^2y}

4) m+n30m4−m+2n30m4\frac{m + n}{30m^4} - \frac{m + 2n}{30m^4}

  1. Combine fractions (denominator 30m430m^4): (m+n)−(m+2n)30m4\frac{(m + n) - (m + 2n)}{30m^4}
  2. Simplify numerator: m+n−m−2n30m4=−n30m4\frac{m + n - m - 2n}{30m^4} = \frac{-n}{30m^4}

5) x+2y15x−x−6y15x\frac{x + 2y}{15x} - \frac{x - 6y}{15x}

  1. Combine fractions (denominator 15x15x): (x+2y)−(x−6y)15x\frac{(x + 2y) - (x - 6y)}{15x}
  2. Simplify numerator: x+2y−x+6y15x=8y15x\frac{x + 2y - x + 6y}{15x} = \frac{8y}{15x}

6) 2÷6+3÷4b2 \div 6 + 3 \div 4b

  1. Rewrite each term as a fraction: 26+34b=13+34b\frac{2}{6} + \frac{3}{4b} = \frac{1}{3} + \frac{3}{4b}
  2. The least common denominator is 12b12b: 4b12b+912b=4b+912b\frac{4b}{12b} + \frac{9}{12b} = \frac{4b + 9}{12b}

7) 4y4−3x4y3\frac{4y}{4} - \frac{3x}{4y^3}

  1. Simplify the first fraction: y−3x4y3y - \frac{3x}{4y^3}
  2. Combine under a common denominator: 4y44y3−3x4y3=4y4−3x4y3\frac{4y^4}{4y^3} - \frac{3x}{4y^3} = \frac{4y^4 - 3x}{4y^3}

8) 5x46−65xy\frac{5x^4}{6} - \frac{6}{5xy}

  1. Combine under a common denominator of 30xy30xy: 25x5y30xy−3630xy\frac{25x^5y}{30xy} - \frac{36}{30xy}
  2. Simplify: 25x5y−3630xy\frac{25x^5y - 36}{30xy}

9) 6a23ab+2b3ab\frac{6a^2}{3ab} + \frac{2b}{3ab}

  1. Combine fractions (denominator 3ab3ab): 6a2+2b3ab\frac{6a^2 + 2b}{3ab}

10) 42y21+26y1\frac{42y^2}{1} + \frac{26y}{1}

  1. Factor common terms: 42y2+26y=2y(21y+13)42y^2 + 26y = 2y(21y + 13)

11) m−24m3+20m2−m+64m3+20m2\frac{m - 2}{4m^3 + 20m^2} - \frac{m + 6}{4m^3 + 20m^2}

  1. Combine fractions: (m−2)−(m+6)4m3+20m2\frac{(m - 2) - (m + 6)}{4m^3 + 20m^2}
  2. Simplify numerator: m−2−m−64m3+20m2=−84m3+20m2\frac{m - 2 - m - 6}{4m^3 + 20m^2} = \frac{-8}{4m^3 + 20m^2}

**12) (\frac{k -

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Math Problem Analysis

Mathematical Concepts

Rational Expressions
Fraction Operations
Common Denominators
Simplification of Algebraic Expressions

Formulas

Addition/Subtraction of Fractions: a/b + c/b = (a+c)/b
Common Denominator Approach: a/b + c/d = (ad + bc)/(bd)

Theorems

Properties of Rational Expressions
Distributive Property

Suitable Grade Level

Grades 9-12

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