Algebra Formula List: Identities, Exponents and Logarithms

What is algebra in maths

Algebra is the part of maths where letters stand for numbers. Instead of saying "some number plus five", you write $x + 5$. That one step lets you state rules that work for every number at once, solve for unknowns and spot patterns. The meaning of algebra in English is close to "the arithmetic of letters"; the word comes from an Arabic title for a book on restoring and balancing equations.

A few words you will see on this page:

• Variable: a letter whose value can change, like x or a.
• Constant: a fixed number, like 7 or $\pi$.
• Term: a product of numbers and letters, such as $3ab$ or $-5x^2$.
• Factor: anything that multiplies to make a term or expression. In $3ab$ the factors are 3, a and b; in $(x + 2)(x - 2)$ the factors are the two brackets.
• Identity: an equation true for all values of the letters. An ordinary equation, like $2x + 1 = 7$, is true only for one value (here $x = 3$).

School algebra builds up through middle school, the standard identities are collected in the class 9 chapter on polynomials, and quadratic equations and arithmetic progressions come in class 10. College courses add linear algebra and abstract algebra, but the identities below are the base for all of it.

Algebra formula chart for squares

The (a+b)2 formula and (a-b)2 formula

Squaring a sum is not the same as adding squares. $(a + b)^2$ means $(a + b)(a + b)$; multiply out and you get $a^2 + ab + ab + b^2$, so the middle term is $2ab$. For $(a - b)^2$ only that middle sign flips. Use them for quick mental squares: $103^2 = (100 + 3)^2 = 10000 + 600 + 9 = 10609$, and $999^2 = (1000 - 1)^2 = 1000000 - 2000 + 1 = 998001$.

The a2-b2 formula

The difference of two squares factorises as $(a + b)(a - b)$. It turns hard subtraction into easy multiplication: $51^2 - 49^2 = (51 + 49)(51 - 49) = 100 \times 2 = 200$. It also splits $x^4 - 16$ into $(x^2 + 4)(x + 2)(x - 2)$ when you apply it twice.

The a2+b2 formula (a^2+b^2)

There is no factor form of $a^2 + b^2$ using real numbers, so the a^2+b^2 formula is written through the other squares: $a^2 + b^2 = (a + b)^2 - 2ab$ or $(a - b)^2 + 2ab$. Use whichever matches the data the question gives you.

The (a+b+c)2 formula

Square each letter and add twice every pair: $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$. Rearranged, it gives the a2+b2+c2 formula: $a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)$.

Cube identities: a plus b whole cube and more

The a + b whole cube formula, (a+b)^3

The a+b whole cube formula has the coefficients 1, 3, 3, 1. Grouping the middle two terms gives the short form $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$, which is the one you will use most. Example: $(2x + 1)^3 = 8x^3 + 12x^2 + 6x + 1$.

The a-b whole cube formula, (a-b)3

Signs alternate: plus, minus, plus, minus. Short form: $(a - b)^3 = a^3 - b^3 - 3ab(a - b)$. For $(2x - 3y)^3$ you get $8x^3 - 36x^2y + 54xy^2 - 27y^3$. Work it once yourself to check the middle coefficients.

The a3+b3 formula, also typed a^3+b^3 formula

The sum of two cubes factorises as $(a + b)(a^2 - ab + b^2)$. Remember the sign pattern "same, opposite, always plus": the first bracket has the same sign as the question, the middle term of the second bracket has the opposite sign, and the last term is plus. Example: $8x^3 + 27 = (2x)^3 + 3^3 = (2x + 3)(4x^2 - 6x + 9)$. From the cube identity you also get $a^3 + b^3 = (a + b)^3 - 3ab(a + b)$, handy when a + b and ab are given.

The a3-b3 formula (a^3-b^3)

The difference of cubes is $(a - b)(a^2 + ab + b^2)$. Example: $x^3 - 64y^3 = (x - 4y)(x^2 + 4xy + 16y^2)$. The related form is $a^3 - b^3 = (a - b)^3 + 3ab(a - b)$.

The a3+b3+c3-3abc formula

This three letter identity is part of class 9 Polynomials and turns up often in class 9 tests.

The most useful consequence: if $a + b + c = 0$, the right side is zero, so

This is what people mean by the a3+b3+c3 formula. It lets you find a sum of three cubes without cubing anything large.

Solved examples using each algebra formula

Laws of exponents

Exponents (powers or indices) say how many times a base is multiplied by itself. The same few laws cover integers, fractions and negative powers, as long as the base is positive or the powers are whole numbers. A zero or negative power needs $a \neq 0$.

With fractional powers, take the root first and then the power; the numbers stay small. For instance $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$, and $8^{-2/3} = \tfrac{1}{4}$. You meet these laws in middle school and again in the class 9 chapter on number systems.

Logarithm laws with examples

A logarithm answers the question "what power?". If $a^x = N$ then $\log_a N = x$, for $a \gt 0$, $a \neq 1$ and $N \gt 0$. Every law below needs m and n positive too, because the log of zero or a negative number is not defined. So $\log_2 32 = 5$ because $2^5 = 32$, and $\log_3 81 = 4$.

Logs are not a separate NCERT chapter, but you meet them in class 11 chemistry (pH) and physics, and in the compound interest formula when you solve for time. Their derivatives and integrals are on the differentiation page.

Common mistakes with algebraic identities

• Writing $(a - b)^2 = a^2 - b^2$. The correct form has $-2ab$ and $+b^2$.
• Mixing up the brackets of $a^3 + b^3$ and $a^3 - b^3$. The second bracket is never $(a + b)^2$ or $(a - b)^2$; its middle term is just $\pm ab$.
• Forgetting to cube the coefficient: $(2x)^3 = 8x^3$, not $2x^3$.
• Applying $a^3 + b^3 + c^3 = 3abc$ without checking that $a + b + c = 0$.
• Adding powers of different bases: $2^3 \cdot 3^2$ cannot be combined into one power.
• Treating $\log(m + n)$ as $\log m + \log n$. The product law needs a product inside the log.

Algebra formula words in Hindi medium books

Studying from a Hindi medium textbook, or typing "a^2+b^2 ka formula", into a search bar? Match these words.

So "a plus b whole square" is read in class as "a plus b ka varg", and the cube as "a plus b ka ghan". For every other chapter's formulas in one place, see the maths formula hub.