There, we introduced the special notation of writing a (literal) number being multiplied more than once, with examples.
We explained the terms : Base, Exponent or Index, Exponential form, Expanded form.
We provided solved examples and problems for practice with answers, to help to understand the conepts covered.
Here, we state the 7 Laws of indices and the two Rules used for solving problems.
We also give the explanations and proofs of the 7 laws.
We provide a few Solved Examples and Problems for Practice with Answers to help to apply the 7 Laws and the 2 Rules in solving problems.
Laws of Exponents (or Indices)
Law 1 of Laws of Exponents :
Law 1: product of powers of the same base: am x an = am + n
Law 2 of Laws of Exponents :
Law 2: power of a power: (am)n = amn
Law 3 of Laws of Exponents :
Law 3: Powers with exponent being negative integer: a-n = 1⁄an
Law 4 of Laws of Exponents :
Law 4: quotient of powers of the same base: am⁄an = am - n
Law 5 of Laws of Exponents :
Law 5: powers with exponent zero: a0 = 1
Law 6 of Laws of Exponents :
Law 6: power of a product: (ab)m = am x bm
Law 7 of Laws of Exponents :
Law 7: power of a quotient: (a⁄b)m = am⁄bm
There are two rules which are useful in solving problems.
Rule 1 of Exponents :
Rule 1: In an Equation involving Exponents, If the BASES (other than 0, 1, -1) of the L.H.S. and the R.H.S. are equal, then, the EXPONENTS should be equal. am = an ⇒ m = n, where a ≠ 0, a ≠ 1 and a ≠ -1.
Rule 2 of Exponents :
Rule 2: In an Equation involving Exponents, If the EXPONENTS (other than 0 ) of the L.H.S. and the R.H.S. are equal, then, the BASES should be equal. am = bm ⇒ a = b, where m ≠ 0.
You have to observe, understand, remember and apply the above Laws in solving various problems.
The above Laws are valid for integral exponents (m and n).
You can apply the above Laws (for integral exponents) by studying the Sets of Solved Examples and doing Exercises, given in the pages for which Links are given atthe bottom of this page.
You can study the Explanations and Proofs of the Laws given below in two Links in the beginning (now) or in the middle or at the end of applying the Laws to various problems and getting thoroughly acquainted with them.
The best way to remember various Algebra Formulas (or Math Formulas) is to apply them to a number of problems. The best way to solve Algebra Problems (or Math Problems) is to remember various formulas.
This is similar to the saying, "if you like a thing, you will concentrate. or if you concentrate on a thing, you will like it."
So, adopt the best way [i.e. trying to memorise the formulas (Laws ) first or going to the problems first or some way in between] that is suitable for you.
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The above 7 Laws and the 2 Rules are valid for Rational Number Exponents also.
The Link to the page which gives Explanation of the Laws for Rational Exponents, and which provides Links to Sets of solved examples and exercises on their application to the Rational Exponents, is given at the bottom.
Explanations and Proofs of the Laws :
For Explanations and Proofs of the above Laws, go to
The following Link takes you to the page which gives Explanation of the Laws for Rational Exponents, and provides Links to Sets of solved examples and exercises on their application to the Rational Exponents.
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