Simplify-1 m 2- n 2 m - n 2 - m 2- mn - n 2 m 3- n 32 a 2-10 a -21 a 2-6 a -7 - a 2-1 a -33 8 - 3-27 y 34 x 2-9 y 24 x 2-5 x-24 x-3 x-8 - x 2-64 x-825 3 x 2-x-2 x 2-7 x-12 - 3 x 2-7 x-6 x 2-46 4 - 2-11 x-616 x 2-97 a 3-275 a 2-16 a-3 - a 2-3 a-925 a 2-18 1-2 x - x 21- x 3 - 1- x - x 21- x

It is known that, a2 b2=a+ba b #160;; #160;a3 b3=a ba2+ab+b2 1 #160;m2 n2m+n2 #215;m2+mn+n2m3 n3= #160;m+nm nm+nm+n #215;m2+mn+n2m nm2+mn+n2= #160 ...

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Byju's Answer

Standard VII

Mathematics

Addition and Subtraction of Algebraic Expressions

Simplify:1 m ...

Question

Simplify:
(1) $\frac{m^{2} - n^{2}}{{(m + n)}^{2}} \times \frac{m^{2} + m n + n^{2}}{m^{3} - n^{3}}$

(2) $\frac{a^{2} + 10 a + 21}{a^{2} + 6 a - 7} \times \frac{a^{2} - 1}{a + 3}$

(3) $\frac{8 x^{3} - 27 y^{3}}{4 x^{2} - 9 y^{2}}$

(4) $\frac{x^{2} - 5 x - 24}{(x + 3) (x + 8)} \times \frac{x^{2} - 64}{{(x - 8)}^{2}}$

(5) $\frac{3 x^{2} - x - 2}{x^{2} - 7 x + 12} \div \frac{3 x^{2} - 7 x - 6}{x^{2} - 4}$

(6) $\frac{4 x^{2} - 11 x + 6}{16 x^{2} - 9}$

(7) $\frac{a^{3} - 27}{5 a^{2} - 16 a + 3} \div \frac{a^{2} + 3 a + 9}{25 a^{2} - 1}$

(8) $\frac{1 - 2 x + x^{2}}{1 - x^{3}} \times \frac{1 + x + x^{2}}{1 + x}$

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Solution

It is known that,
$a^{2} - b^{2} = (a + b) (a - b); a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
$(1) \frac{m^{2} - n^{2}}{{(m + n)}^{2}} \times \frac{m^{2} + m n + n^{2}}{m^{3} - n^{3}} = \frac{(m + n) (m - n)}{(m + n) (m + n)} \times \frac{(m^{2} + m n + n^{2})}{(m - n) (m^{2} + m n + n^{2})} = \frac{1}{m + n}$
$(2) \frac{a^{2} + 10 a + 21}{a^{2} + 6 a - 7} \times \frac{a^{2} - 1}{a + 3} = \frac{a^{2} + 7 a + 3 a + 21}{a^{2} + 7 a - a - 7} \times \frac{(a + 1) (a - 1)}{(a + 3)} = \frac{a (a + 7) + 3 (a + 7)}{a (a + 7) - 1 (a + 7)} \times \frac{(a + 1) (a - 1)}{(a + 3)} = \frac{(a + 7) (a + 3)}{(a - 1) (a + 7)} \times \frac{(a + 1) (a - 1)}{(a + 3)} = (a + 1)$
$(3) \frac{8 x^{3} - 27 y^{3}}{4 x^{2} - 9 y^{2}} = \frac{{(2 x)}^{3} - {(3 y)}^{3}}{{(2 x)}^{2} - {(3 y)}^{2}} = \frac{(2 x - 3 y) [{(2 x)}^{2} + (2 x) \times (3 y) + {(3 y)}^{2}]}{(2 x + 3 y) (2 x - 3 y)} = \frac{(2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})}{(2 x + 3 y) (2 x - 3 y)} = \frac{4 x^{2} + 6 x y + 9 y^{2}}{(2 x + 3 y)}$
$(4) \frac{x^{2} - 5 x - 24}{(x + 3) (x + 8)} \times \frac{x^{2} - 64}{{(x - 8)}^{2}} = \frac{x^{2} - 8 x + 3 x - 24}{(x + 3) (x + 8)} \times \frac{{(x)}^{2} - {(8)}^{2}}{{(x - 8)}^{2}} = \frac{x (x - 8) + 3 (x - 8)}{(x + 3) (x + 8)} \times \frac{(x + 8) (x - 8)}{(x - 8) (x - 8)} = \frac{(x + 3) (x - 8)}{(x + 3) (x + 8)} \times \frac{(x + 8) (x - 8)}{(x - 8) (x - 8)} = 1$
$(5) \frac{3 x^{2} - x - 2}{x^{2} - 7 x + 12} \div \frac{3 x^{2} - 7 x - 6}{x^{2} - 4} = \frac{3 x^{2} - 3 x + 2 x - 2}{x^{2} - 4 x - 3 x + 12} \div \frac{3 x^{2} - 9 x + 2 x - 6}{{(x)}^{2} - {(2)}^{2}} = \frac{3 x (x - 1) + 2 (x - 1)}{x (x - 4) - 3 (x - 4)} \div \frac{3 x (x - 3) + 2 (x - 3)}{(x + 2) (x - 2)} = \frac{(3 x + 2) (x - 1)}{(x - 3) (x - 4)} \div \frac{(3 x + 2) (x - 3)}{(x + 2) (x - 2)} = \frac{(3 x + 2) (x - 1)}{(x - 3) (x - 4)} \times \frac{(x + 2) (x - 2)}{(3 x + 2) (x - 3)} = \frac{(x - 1) (x + 2) (x - 2)}{(x - 4) (x - 3) (x - 3)}$
$(6) \frac{4 x^{2} - 11 x + 6}{16 x^{2} - 9} = \frac{4 x^{2} - 8 x - 3 x + 6}{{(4 x)}^{2} - {(3)}^{2}} = \frac{4 x (x - 2) - 3 (x - 2)}{(4 x + 3) (4 x - 3)} = \frac{(4 x - 3) (x - 2)}{(4 x + 3) (4 x - 3)} = \frac{x - 2}{4 x + 3}$
$(7) \frac{a^{3} - 27}{5 a^{2} - 16 a + 3} \div \frac{a^{2} + 3 a + 9}{25 a^{2} - 1} = \frac{{(a)}^{3} - {(3)}^{3}}{5 a^{2} - 15 a - a + 3} \div \frac{a^{2} + 3 a + 9}{{(5 a)}^{2} - {(1)}^{2}} = \frac{(a - 3) \{{(a)}^{2} + (a) \times (3) + {(3)}^{2}\}}{5 a (a - 3) - 1 (a - 3)} \div \frac{a^{2} + 3 a + 9}{(5 a + 1) (5 a - 1)} = \frac{(a - 3) (a^{2} + 3 a + 9)}{(5 a - 1) (a - 3)} \div \frac{(a^{2} + 3 a + 9)}{(5 a + 1) (5 a - 1)} = \frac{(a - 3) (a^{2} + 3 a + 9)}{(5 a - 1) (a - 3)} \times \frac{(5 a + 1) (5 a - 1)}{(a^{2} + 3 a + 9)} = 5 a + 1$
$(8) \frac{1 - 2 x + x^{2}}{1 - x^{3}} \times \frac{1 + x + x^{2}}{1 + x} = \frac{1 - x - x + x^{2}}{{(1)}^{3} - {(x)}^{3}} \times \frac{1 + x + x^{2}}{1 + x} = \frac{1 (1 - x) - x (1 - x)}{(1 - x) \{{(1)}^{2} + (1) \times (x) + {(x)}^{2}\}} \times \frac{(1 + x + x^{2})}{1 + x} = \frac{(1 - x) (1 - x)}{(1 - x) (1 + x + x^{2})} \times \frac{(1 + x + x^{2})}{(1 + x)} = \frac{1 - x}{1 + x}$

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Similar questions

Q. Simplify :

\frac{a^{3} - 27}{5 a^{2} - 16 a + 3} \div \frac{a^{2} + 3 a + 9}{25 a^{2} - 1}

Q. If

(1 + x + x^{2}) (1 - \frac{x}{1!} + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \dots) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + . . .

then,

Q. Simplify :

\frac{3 x^{2} - x - 2}{x^{2} - 7 x + 12} \div \frac{3 x^{2} - 7 x - 6}{x^{2} - 4}

Q. Divide and write the quotient and the remainder.
(1) (y² + 10y + 24) ÷ (y + 4)
(2) (p² + 7p − 5) ÷ (p + 3)
(3) (3x + 2x² + 4x³) ÷ (x − 4)
(4) (2m³ + m² + m + 9) ÷ (2m − 1)
(5) (3x − 3x² − 12 + x⁴ + x³) ÷ (2 + x²)
(6) (a⁴ − a³ + a² − a + 1) ÷ (a³ − 2)
(7) (4x⁴ − 5x³ − 7x + 1) ÷ (4x − 1)

Q. Which of the following are quadratic equations?

(i) x² + 6x − 4 = 0
(ii)

\sqrt{3} x^{2} - 2 x + \frac{1}{2} = 0

(iii)

x^{2} + \frac{1}{x^{2}} = 5

(iv)

x - \frac{3}{x} = x^{2}

(v)

2 x^{2} - \sqrt{3 x} + 9 = 0

(vi)

x^{2} - 2 x - \sqrt{x} - 5 = 0

(vii) 3x² − 5x + 9 = x² − 7x + 3
(viii)

x + \frac{1}{x} = 1

(ix) x² − 3x = 0
(x)

{(x + \frac{1}{x})}^{2} = 3 (x + \frac{1}{x}) + 4

(xi) (2x + 1) (3x + 2) = 6(x − 1) (x − 2)
(xii)

x + \frac{1}{x} = x^{2}, x \neq 0

(xiii) 16x² − 3 = (2x + 5) (5x − 3)
(xiv) (x + 2)³ = x³ − 4
(xv) x(x + 1) + 8 = (x + 2) (x − 2)

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