Algebraic Formulae (66)

Algebra: Core Concepts and Formulas

Algebra is one of the most basic parts of mathematics. Algebra deals with variables and numbers.

Although Algebra is basic mathematics, it deals with a large number of formulas which relate two or more variables and numbers with each other.

To help you, here is a list of Algebraic formulas.

Basic Algebraic Formulas

The algebraic formulas involving basic relations between two and three variables are:

1. 1.
   \[a^2-b^2 = (a+b)(a-b)\]
2. 2.
   \[a^2+b^2 = (a+b)^2-2ab = (a-b)^2+2ab\]
3. 3.
   \[(a+b)^2 = a^2+2ab+b^2 = (a-b)^2+4ab\]
4. 4.
   \[(a-b)^2 = a^2-2ab+b^2 = (a+b)^2-4ab\]
5. 5.
   \[a^3+b^3 = (a+b)(a^2-ab+b^2) = (a+b)^3-3ab(a+b)\]
6. 6.
   \[a^3-b^3 = (a-b)(a^2+ab+b^2) = (a-b)^3+3ab(a-b)\]
7. 7.
   \[(a+b)^3 = a^3+3a^2b+3ab^2+b^3 = a^3+b^3+3ab(a+b)\]
8. 8.
   \[(a-b)^3 = a^3-3a^2b+3ab^2-b^3 = a^3-b^3-3ab(a-b)\]
9. 9.
   \[(x+a)(x+b) = x^2+x(a+b)+ab\]
10. 10.
    \[(x-a)(x+b) = x^2+x(b-a)-ab\]
11. 11.
    \[(x-a)(x-b) = x^2-x(a+b)+ab\]
12. 12.
    \[(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ac\]
13. 13.
    \[(a+b+c)^3 = a^3+b^3+c^3+3(a+b)(b+c)(c+a)\]
14. 14.
    \[a^4-b^4 = (a-b)(a+b)(a^2+b^2)\]
15. 15.
    \[a^6-b^6 = (a+b)(a^2-ab+b^2)(a-b)(a^2+ab+b^2)\]
16. 16.
    \[a^6+b^6 = (a^2+b^2)(a^4-a^2b^2+b^4)\]
17. 17.
    \[a^4+a^2b^2+b^4 = (a^2-ab+b^2)(a^2+ab+b^2)\]
18. 18.
    \[(a-b-c)^2 = a^2+b^2+c^2-2ab+2bc-2ac\]
19. 19.
    \[a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ac)\]

Indices Formulas

The formulas involving relations between variables and their powers or powers and indices are:

1. 1.
   \[x^m \times x^n = x^{m+n}\] \[x^m \times x^n \times \ldots \times x^p = x^{m+n+ \ldots +p}\]
2. 2.
   \[x^m \div x^n = x^{m-n}\] \[x^m \div x^n \div \ldots \div x^p = x^{m-n- \ldots -p}\]
3. 3.
   \[(x^m)^n = x^{m \times n}\] \[((x^m)^n)^o = x^{m \times n \times o}\]
4. 4.
   \[x^0 = 1 \quad (\text{for } x \neq 0)\]
5. 5.
   \[x^{-m} = \dfrac{1}{x^m}\] \[x^{m} = \dfrac{1}{x^{-m}}\]
6. 6.
   \[x^{\frac{m}{n}} = \sqrt[n]{x^m}\]
7. 7.
   \[\left( \dfrac{x^a}{y^b} \right)^c = \dfrac{x^{ac}}{y^{bc}}\]
8. 8.
   \[\dfrac{x^m}{y^m} = \left( \dfrac{x}{y} \right)^m\]
9. 9.
   \[\sqrt[m]{\dfrac{x^a}{y^b}} = \dfrac{x^{\frac{a}{m}}}{y^{\frac{b}{m}}}\]
10. 10.
    \[x^{\frac{p}{q}} = \sqrt[q]{x^p} = \left(\sqrt[q]{x}\right)^p\]
11. 11.
    \[\sqrt[m]{\dfrac{x}{y}} = \dfrac{\sqrt[m]{x}}{\sqrt[m]{y}}\]
12. 12.
    \[\sqrt{a} \times \sqrt {b} = \sqrt{a \times b}\]

    Provided that a, b are not negative numbers.
13. 13.
    \[\text{If } a^x = a^y, \text{ then } x=y\]

    Provided that: \(0 < a \text{ and } a \neq 1\)
14. 14.
    \[\text{If } a^x = b^x, \text{ then } a=b\]

    Provided that: \(0 < a, b \text{ and } x \neq 0\) (Original said a,b !=1, but x!=0 is more critical here if a,b can be anything positive)

    Alternative simpler condition: \(x \neq 0 \text{ and } a, b > 0\).
    If strictly following original LaTeX: \(0 < a, b \text{ and } a \neq 1, b \neq 1 \text{ (and implying } x \neq 0)\)