(i) \(\displaystyle a^{m} . a^{n} = a^{m+n}\)
(ii) \(\displaystyle \frac{ a^{m}}{a^{n}} = a^{m- n}\)
(iii) \(\displaystyle (a^{m})^{n} = a^{mn}\)
(iv) \(\displaystyle a^{-n} = \frac{1}{a^{n}} \)
(v) \(\displaystyle \sqrt[n]{a^{m}}= \displaystyle a^{\frac{m}{n}}\)
(vii)\(\displaystyle (ab)^{m} = a^{m}. b^{m}\).
(viii)\(\displaystyle \left (\frac{a}{b} \right )^{m} = \frac{a^{m}}{b^{m}}\)
(ix) If \(\displaystyle a^{m} = b^{m} (m ≠ 0), \;\;then\;\; a = b.\)
(x) If \(\displaystyle a^{m}= a^{n}\;\; then\;\; m = n.\)
● Quadratic Equation Formulas :
Quadratic equation is of form \(ax² + bx + c = 0 .\)....*
(i) Roots of the quadratic equation are \(\alpha, \beta = \displaystyle \frac{-b ± \sqrt{(b^{2} – 4ac)}}{2a}\).
(ii) If \(\alpha\) and \(\beta\) be the roots of the quadratic equation then,
sum of its roots \(= \alpha+\beta = \displaystyle \frac{-b}{a}\) or we can say \(\displaystyle \frac{- (\text{coefficient of x})}{\text{(coefficient of \(x^2\) )}}\)
and product of its roots \(= \alpha.\beta = \displaystyle \frac{c}{a}\) or we can say \(\displaystyle \frac{\text{Constant term}} {\text{(Coefficient of \(x^{2}\))}}\).
(iii) The quadratic equation with both real roots are \(\alpha\) and \(\beta\) is
\(x^2 - (\alpha + \beta)x + \alpha.\beta= 0\)
i.e. , \(x^2 - (\text{sum of the roots}) x + \text{product of the roots} = 0.\)
(iv) The expression \((b^2 - 4ac)\) is called the discriminant of quadratic equation. Denoted by \(D\)
(v) If \(a, b, c\) are real and rational then the roots of quadratic equation are
(a) Real and equal when \(b^2 - 4ac = 0\).
(b) Real and distinct when \(b^2 - 4ac > 0\).
(c) Imaginary when \(b^2 - 4ac < 0\).
(d)Rational when \(b^2- 4ac\) is a perfect square and
(e) Irrational when \(b^2 - 4ac\) is not a perfect square.
(vi) If \(\alpha + i\beta\) be one root of quadratic equation then its other root will be complex conjugate of this i.e \(\alpha - i\beta\) .
(vii) If \(\alpha +\sqrt{\beta}\) be one root of equation quadratic equation then its other root will be conjugate irrational quantity \(\alpha - \sqrt{\beta}\) (a, b, c are rational).
● Arithmetical Progression (A.P.):
(i) The general form of an A. P. is \(a, a + d, a + 2d, a + 3d,.....\)
where a is the first term and d, the common difference of the A.P.
(ii) The nth term of the above A.P. is \(a_{n} = a + (n - 1)d.\)
(iii) The sum of first n terns of the above A.P. is \(\displaystyle S_{n} = \frac{n}{2} (a + l)\) (where l is last term )or, \(\displaystyle S_{n} = \frac{n}{2} [2a + (n - 1) d]\)
(iv) The arithmetic mean between two given numbers a and b is \(\displaystyle \frac{(a + b)}{2}\).
(v) \(1 + 2 + 3 + ...... + n =\displaystyle \frac{ n(n + 1)}{2}\).
(vi) \(1^{2} + 2^{2} + 3^{2} +… + n^{2} = \displaystyle\frac{n(n+ 1)(2n+ 1)}{6}.\)
(vii) \(1^{3} + 2^{3} + 3^{3} + . . . . + n^{3}= \displaystyle \frac{{n^{2}(n + 1)^{2}}}{4 }\).
● Geometrical Progression (G.P.) :
(i) The terms G.P. is \(a, ar, ar^{2}, ar^{3}, . . . . .\) where a is the first term and r, the common ratio of the G.P.
(ii) The nth term given G.P. is \(t_{r} = a.r^{n-1} \) .
(iii) The sum of first n terms of the above G.P. is \(S_{n} =\displaystyle \frac{a . (1 - r^{n})}{(1 – r)}\)when \(-1 < r < 1\)
or, \(S _{n}= \displaystyle \frac{a .(r^{n} – 1)}{(r – 1) }\) when \(r > 1\) or \(r < -1\)
(iv) \(a + ar + ar^{2} +........+\infty = \displaystyle \frac{a}{1 – r}\) where \(-1 < r < 1\).
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