Lesson BASICS - Laws of Indices
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<b>Introduction</b> The manipulation of powers, or indices or exponents is a very crucial underlying skill to have in algebra. In essence there are just 3 laws and from those we can derive 3 other interesting/useful rules. <b>Basic Terminology</b> What does {{{x^3}}} or {{{x^2y^3}}} mean? Well, they are just short-hand notation for the following: {{{x^3}}} means x.x.x {{{x^2y^3}}} means x.x.y.y.y where the . symbol means "multiplied by" Remember that you cannot multiply letters. It does not make sense. All we can do is keep "x multiplied by x" as that. However, we write it in a neater form, namely {{{x^2}}}. <b>The 3 Laws of Indices</b> <b>1.</b> {{{ (x^m)(x^n) = x^(m+n) }}} eg {{{(x^3)(x^4) }}} is the same thing as {{{ x^7 }}}. We can see the reason, if we write out the x's fully: {{{(x^3)(x^4) }}} means x.x.x . x.x.x.x which is just {{{x^7}}}. <b>2.</b> {{{ (x^m)/(x^n) = x^(m-n)}}} eg. {{{ x^7/x^4 }}} is the same thing as {{{x^3}}}. Again, we can see the reason for this if we write out the x's fully: {{{ x^7/x^4 }}} means (x.x.x.x.x.x.x)/(x.x.x.x) which we can cancel down to x.x.x which is just {{{x^3}}}. <b>3.</b> {{{ (x^m)^n = x^(mn) }}} eg {{{ (x^2)^3 }}} is the same thing as {{{x^6}}}. We can see the reason for this by writing out the question fully, again: {{{ (x^2)^3 }}} means (x.x).(x.x).(x.x) which is {{{x^6}}}. --------------------------------------------------------------------------------- <b>The Additional 3 Rules of Indices</b> <b>4.</b> when n=m, we get, from Rule 2, that {{{ (x^m)/(x^m) = x^(m-m) }}} --> {{{ 1 = x^0}}} --> {{{ x^0 = 1 }}} --> <b> Anything to the power zero is 1</b> eg {{{ (x^2y^5)^0 }}} is 1 <b>5.</b> when n=-m, we get, from Rule 1, that {{{ (x^m)x^(-m) = x^(m+(-m)) }}} --> {{{ (x^m)x^(-m) = x^0 }}} --> {{{ (x^m)x^(-m) = 1 }}} so, {{{ x^(-m) = 1/x^m }}} --> This is the physical meaning of negative powers. eg. {{{ a^(-3) }}} means {{{ 1/a^3 }}}. eg {{{ 5^(-2) }}} means {{{ 1/5^2 }}} which is {{{1/25}}} <b>6.</b> when n=1/m, we get, from Rule 3, that {{{ (x^m)^(1/m) = x^((m)(1/m)) }}} --> {{{ (x^m)^(1/m) = x^1 }}} so, {{{ (x^m)^(1/m) = x }}} --> This is the physical meaning of fractional powers. The rule above says if you raise x to a power "m" and then raise to power "1/m", we end up back with x --> m and 1/m are opposite operations. So, if we raise to power 2, the opposite of that is square root. So power "1/2" means a square root. if we raise to power 3, the opposite of that is cube root. So power "1/3" means a cube root. In general, if we raise to power m, the opposite of that is mth root. So power "1/m" means a mth root eg {{{ 125^(1/3) }}} means the cube root of 125, which is 5 since 5.5.5 = 125 --------------------------------------------------------------------------------- <b>Examples</b> Calculate the folowing: 1. {{{ 9^0 }}} is 1 2. {{{ 2^(-5) }}} is {{{ 1/2^5 }}} which is {{{ 1/32 }}}. 3. {{{ 81^(1/4) }}} is 3 since 3.3.3.3 is 81 4. {{{ 343^(-1/3) }}} is {{{ 1/343^(1/3) }}}. The answer is {{{ 1/7}}} since 7.7.7 is 343.
