Fundamental Concepts of Algebra — Exercise 12(E)

Divide:

$3a$ by $a$

Answer

Solving,

$\Rightarrow \dfrac{3a}{a}$

$\Rightarrow 3$

Hence, $\bm{\dfrac{3a}{a} = 3}$.

Divide:

$15x$ by $3x$

Answer

Solving,

$\Rightarrow \dfrac{15x}{3x}$

$\Rightarrow 5$

Hence, $\bm{\dfrac{15x}{3x} = 5}$.

Divide:

$16m$ by 4

Answer

Solving,

$\Rightarrow \dfrac{16m}{4}$

$\Rightarrow 4m$

Hence, $\bm{\dfrac{16m}{4} = 4m}$.

Divide:

$20x^2$ by $5x$

Answer

Solving,

$\Rightarrow \dfrac{20x^2}{5x} \\[1em] \Rightarrow 4x^{2-1} \\[1em] \Rightarrow 4x$

Hence, $\bm{\dfrac{20x^2}{5x} = 4x}$.

Divide:

$30p^2$ by $10p^2$

Answer

Solving,

$\Rightarrow \dfrac{30p^2}{10p^2}$

$\Rightarrow 3$

Hence, $\bm{\dfrac{30p^2}{10p^2} = 3}$.

Divide:

$14a^3b^3$ by $2a^2$

Answer

Solving,

$\Rightarrow \dfrac{14a^3b^3}{2a^2} \\[1em] \Rightarrow 7a^{3-2}b^3 \\[1em] \Rightarrow 7ab^3$

Hence, $\bm{\dfrac{14a^3b^3}{2a^2} = 7ab^3}$.

Divide:

$18pqr^2$ by $3pq$

Answer

Solving,

$\Rightarrow \dfrac{18pqr^2}{3pq}$

$\Rightarrow 6r^2$

Hence, $\bm{\dfrac{18pqr^2}{3pq} = 6r^2}$.

Divide:

100 by $50b$

Answer

Solving,

$\Rightarrow \dfrac{100}{50b} = \dfrac{2}{b}$

Hence, $\bm{\dfrac{100}{50b} = \dfrac{2}{b}}$.

Simplify:

$2x^5 \div x^2$

Answer

Solving,

$\Rightarrow \dfrac{2x^5}{x^2} \\[1em] \Rightarrow 2x^{5-2} \\[1em] \Rightarrow 2x^3$

Hence, $\bm{2x^5 \div x^2 = 2x^3}$.

Simplify:

$6a^8 \div 3a^3$

Answer

Solving,

$\Rightarrow \dfrac{6a^8}{3a^3} \\[1em] \Rightarrow 2a^{8-3} \\[1em] \Rightarrow 2a^5$

Hence, $\bm{6a^8 \div 3a^3 = 2a^5}$.

Simplify:

$20xy \div -5xy$

Answer

Solving,

$\Rightarrow \dfrac{20xy}{-5xy}$

$\Rightarrow -4$

Hence, $\bm{20xy \div -5xy = -4}$.

Simplify:

$-24a^2b^2c^2 \div 6ab$

Answer

Solving,

$\Rightarrow \dfrac{-24a^2b^2c^2}{6ab} \\[1em] \Rightarrow -4a^{2-1}b^{2-1}c^2 \\[1em] \Rightarrow -4abc^2$

Hence, $\bm{-24a^2b^2c^2 \div 6ab = -4abc^2}$.

Simplify:

$-5x^2y \div xy^2$

Answer

Solving,

$\Rightarrow \dfrac{-5x^2y}{xy^2} \\[1em] \Rightarrow \dfrac{-5x^{2-1}}{y^{2-1}}\\[1em] \Rightarrow \dfrac{-5x}{y}$

Hence, $\bm{-5x^2y \div xy^2 = \dfrac{-5x}{y}}$.

Simplify:

$40p^3q^4r^5 \div 10p^3q$

Answer

Solving,

$\Rightarrow \dfrac{40p^3q^4r^5}{10p^3q} \\[1em] \Rightarrow 4q^{4-1}r^5 \\[1em] \Rightarrow 4q^3r^5$

Hence, $\bm{40p^3q^4r^5 \div 10p^3q = 4q^3r^5}$.

Simplify:

$-64x^4y^3z \div 4x^3y^2z$

Answer

Solving,

$\Rightarrow \dfrac{-64x^4y^3z}{4x^3y^2z} \\[1em] \Rightarrow -16x^{4-3}y^{3-2} \\[1em] \Rightarrow -16xy$

Hence, $\bm{-64x^4y^3z \div 4x^3y^2z = -16xy}$.

Simplify:

$35xy^5 \div 7x^2y^4$

Answer

Solving,

$\Rightarrow \dfrac{35xy^5}{7x^2y^4} \\[1em] \Rightarrow \dfrac{5y^{5-4}}{x^{2-1}} \\[1em] \Rightarrow \dfrac{5y}{x}$

Hence, $\bm{35xy^5 \div 7x^2y^4 = \dfrac{5y}{x}}$.

Divide:

$-\dfrac{3m}{4}$ by $2m$

Answer

Solving,

$\Rightarrow -\dfrac{3m}{4} \div 2m \\[1em] \Rightarrow -\dfrac{3m}{4} \times \dfrac{1}{2m} \\[1em] \Rightarrow \dfrac{-3m}{8m} \\[1em] \Rightarrow \dfrac{-3}{8}$

Hence, $\bm{-\dfrac{3m}{4} \div 2m = \dfrac{-3}{8}}$.

Divide:

$-15p^6q^8$ by $-5p^6q^7$

Answer

Solving,

$\Rightarrow \dfrac{-15p^6q^8}{-5p^6q^7} \\[1em] \Rightarrow 3q^{8-7} \\[1em] \Rightarrow 3q$

Hence, $\bm{\dfrac{-15p^6q^8}{-5p^6q^7} = 3q}$.

Divide:

$-21m^5n^7$ by $14m^2n^2$

Answer

Solving,

$\Rightarrow \dfrac{-21m^5n^7}{14m^2n^2} \\[1em] \Rightarrow \dfrac{-3}{2}m^{5-2}n^{7-2} \\[1em] \Rightarrow \dfrac{-3}{2}m^3n^5$

Hence, $\bm{\dfrac{-21m^5n^7}{14m^2n^2} = \dfrac{-3}{2}m^3n^5}$.

Divide:

$36a^4x^5y^6$ by $4x^2a^3y^2$

Answer

Solving,

$\Rightarrow \dfrac{36a^4x^5y^6}{4x^2a^3y^2} \\[1em] \Rightarrow 9a^{4-3}x^{5-2}y^{6-2} \\[1em] \Rightarrow 9ax^3y^4$

Hence, $\bm{\dfrac{36a^4x^5y^6}{4x^2a^3y^2} = 9ax^3y^4}$.

Divide:

$20x^3a^6$ by $5xy$

Answer

Solving,

$\Rightarrow \dfrac{20x^3a^6}{5xy} \\[1em] \Rightarrow \dfrac{4x^{3-1}a^6}{y} \\[1em] \Rightarrow \dfrac{4x^2a^6}{y}$

Hence, $\bm{\dfrac{20x^3a^6}{5xy} = \dfrac{4x^2a^6}{y}}$.

Divide:

$\dfrac{28a^2b^3}{c^2}$ by $4abc$

Answer

Solving,

$\Rightarrow \dfrac{28a^2b^3}{c^2} \div 4abc \\[1em] \Rightarrow \dfrac{28a^2b^3}{c^2} \times \dfrac{1}{4abc} \\[1em] \Rightarrow \dfrac{28a^2b^3}{4abc^3} \\[1em] \Rightarrow \dfrac{7ab^2}{c^3}$

Hence, $\bm{\dfrac{28a^2b^3}{c^2} \div 4abc = \dfrac{7ab^2}{c^3}}$.

Divide:

$\dfrac{2a^2}{9b^2}$ by $\dfrac{3b}{2a}$

Answer

Solving,

$\Rightarrow \dfrac{2a^2}{9b^2} \div \dfrac{3b}{2a} \\[1em] \Rightarrow \dfrac{2a^2}{9b^2} \times \dfrac{2a}{3b} \\[1em] \Rightarrow \dfrac{4a^3}{27b^3}$

Hence, $\bm{\dfrac{2a^2}{9b^2} \div \dfrac{3b}{2a} = \dfrac{4a^3}{27b^3}}$.

Divide:

$\dfrac{-5.5x^2}{y}$ by $\dfrac{11x}{y}$

Answer

Solving,

$\Rightarrow \dfrac{-5.5x^2}{y} \div \dfrac{11x}{y} \\[1em] \Rightarrow \dfrac{-5.5x^2}{y} \times \dfrac{y}{11x} \\[1em] \Rightarrow \dfrac{-5.5x^2y}{11xy} \\[1em] \Rightarrow \dfrac{-5.5x}{11} \\[1em] \Rightarrow -\dfrac{x}{2} \\[1em] \Rightarrow -0.5x$

Hence, $\bm{\dfrac{-5.5x^2}{y} \div \dfrac{11x}{y} = -\dfrac{x}{2} = -0.5x}$.

Divide:

$\dfrac{64x^2y^2}{z^2}$ by $\dfrac{8xy}{z}$

Answer

Solving,

$\Rightarrow \dfrac{64x^2y^2}{z^2} \div \dfrac{8xy}{z} \\[1em] \Rightarrow \dfrac{64x^2y^2}{z^2} \times \dfrac{z}{8xy} \\[1em] \Rightarrow \dfrac{64x^2y^2z}{8xyz^2} \\[1em] \Rightarrow \dfrac{8xy}{z}$

Hence, $\bm{\dfrac{64x^2y^2}{z^2} \div \dfrac{8xy}{z} = \dfrac{8xy}{z}}$.

Simplify:

$\dfrac{-15m^5n^2}{-3m^5}$

Answer

Solving,

$\Rightarrow \dfrac{-15m^5n^2}{-3m^5}$

$\Rightarrow 5n^2$

Hence, $\bm{\dfrac{-15m^5n^2}{-3m^5} = 5n^2}$.

Simplify:

$\dfrac{35x^4y^2}{-15x^2y^2}$

Answer

Solving,

$\Rightarrow \dfrac{35x^4y^2}{-15x^2y^2} \\[1em] \Rightarrow \dfrac{-7x^{4-2}}{3} \\[1em] \Rightarrow \dfrac{-7x^2}{3}$

Hence, $\bm{\dfrac{35x^4y^2}{-15x^2y^2} = \dfrac{-7x^2}{3}}$.

Simplify:

$\dfrac{-24x^6y^2}{6x^6y}$

Answer

Solving,

$\Rightarrow \dfrac{-24x^6y^2}{6x^6y} \\[1em] \Rightarrow -4y^{2-1} \\[1em] \Rightarrow -4y$

Hence, $\bm{\dfrac{-24x^6y^2}{6x^6y} = -4y}$.

Divide:

$9x^3 - 6x^2$ by $3x$

Answer

Solving,

$\Rightarrow \dfrac{9x^3 - 6x^2}{3x} \\[1em] \Rightarrow \dfrac{9x^3}{3x} - \dfrac{6x^2}{3x} \\[1em] \Rightarrow 3x^2 - 2x$

Hence, $\bm{\dfrac{9x^3 - 6x^2}{3x} = 3x^2 - 2x}$.

Divide:

$6m^2 - 16m^3 + 10m^4$ by $-2m$

Answer

Solving,

$\Rightarrow \dfrac{6m^2 - 16m^3 + 10m^4}{-2m} \\[1em] \Rightarrow \dfrac{6m^2}{-2m} - \dfrac{16m^3}{-2m} + \dfrac{10m^4}{-2m} \\[1em] \Rightarrow -3m - (-8m^2) + (-5m^3) \\[1em] \Rightarrow -3m + 8m^2 - 5m^3$

Hence, $\bm{\dfrac{6m^2 - 16m^3 + 10m^4}{-2m} = -3m + 8m^2 - 5m^3}$.

Divide:

$15x^3y^2 + 25x^2y^3 - 36x^4y^4$ by $5x^2y^2$

Answer

Solving,

$\Rightarrow \dfrac{15x^3y^2 + 25x^2y^3 - 36x^4y^4}{5x^2y^2} \\[1em] \Rightarrow \dfrac{15x^3y^2}{5x^2y^2} + \dfrac{25x^2y^3}{5x^2y^2} - \dfrac{36x^4y^4}{5x^2y^2} \\[1em] \Rightarrow 3x + 5y - \dfrac{36}{5}x^2y^2$

Hence, $\bm{\dfrac{15x^3y^2 + 25x^2y^3 - 36x^4y^4}{5x^2y^2} = 3x + 5y - \dfrac{36}{5}x^2y^2}$.

Divide:

$36a^3x^5 - 24a^4x^4 + 18a^5x^3$ by $-6a^3x^3$

Answer

Solving,

$\Rightarrow \dfrac{36a^3x^5 - 24a^4x^4 + 18a^5x^3}{-6a^3x^3} \\[1em] \Rightarrow \dfrac{36a^3x^5}{-6a^3x^3} - \dfrac{24a^4x^4}{-6a^3x^3} + \dfrac{18a^5x^3}{-6a^3x^3} \\[1em] \Rightarrow -6x^2 - (-4ax) + (-3a^2) \\[1em] \Rightarrow -6x^2 + 4ax - 3a^2$

Hence, $\bm{\dfrac{36a^3x^5 - 24a^4x^4 + 18a^5x^3}{-6a^3x^3} = -6x^2 + 4ax - 3a^2}$.