To increase an index expression to a power, multiply the indexes. Example: [begin{align*} left(5^{2}right)^{3} & =5^{2}times5^{2}times5^{2} & =5^{2+2+2}quadtextrm{using the first index law} & =5^{6} end{align*}] Therefore (left(5^{2}right)^{3}=5^{2times3}=5^{6}.) General: To multiply index expressions, add indexes. Example: [begin{align*} 2^{3}times2^{2} & =left(2times2times2right)timesleft(2times2right) & =2times2times2times2times2times2 & =2^{5} end{align*}] Therefore (2^{3}times2^{2}=2^{3+2}=2^{5}). Similarly, here we have the cubic root of X squared, so the square is at the top of our new index, the cube-shaped root is at the bottom of our new index, so we give an answer of X to the two out of three or X to two-thirds. Now let`s look at the fourth law of indices. Note here that we have parentheses with an index inside and an index outside. So when we have this situation, we multiply the indices, we don`t add them up, we multiply them, so here X to A all the inner parentheses with an index of B outside the parentheses gives us an answer of X at a time A B. For example, four dice in brackets to the power of two would give us four times four multiplied by four times four, that`s the long way to go, which gives us an answer of four to six. But when you apply the law, you multiply the clues, in other words, four dice to the power of two is four to three times two, that is to say four to six, as you can see a much faster way to solve this problem.
Any number other than 0 whose index is 0 is always equal to 1, regardless of the value of the database. Note that index expressions can only be multiplied or split if they have the same base. So far, we have only considered expressions where each index is a positive integer11 integers are called integers and positive integers are called positive integers. Index laws also apply if the index is zero, negative, or a fraction (fractional indices are treated in another module). Finally, consider square roots and cubic roots as clues. Keep a few things in mind here; that the index in the root directory is the numerator of our new index and that the index outside the root directory is the denominator of our new index. Let`s illustrate this with an example, here we have the square root of X which is identical to X to the power of one in square roots, and we could also put a two outside the square root and use the rule above which gives us an answer from X to the one out of two, or X half, so that the square root of X is the same as half X. Note that terms with different bases must be considered separately when using index laws, for example ((2a^{3}b^{2})^{4}=2^{4}a^{12}b^{8}) What is index notation? If a number like 16 is written as 42 (meaning 4 x 4), say it is written exponentially or in index notation. If the index is negative, set it above 1 and turn it over (write its reciprocal) to make it positive.
Let`s look at a few examples; X to the power of four, well, X is the base, four is the index, so it`s X multiplied by itself four times, X times X times X times X. Here we have A squared times B to the power of five. Note here that there are two different bases, A and B, so by multiplying them, A square is A times A and B to the power of five is B times B times B times B. B. Finally, we rolled four times M times cubic times N, so we leave the only four and multiply the M three times, M times M times, and the N is also multiplied three times, N times N times, again the bases are different. This module introduces rules for multiplying and dividing expressions using index notation. For example, how to simplify expressions like (4a^{3}btimes3ab^{5}) or (9a^{3}b^{2}cdiv3ab^{5}). We do not consider fractional indices that are processed in another module.
The index plural is index. If the subscript is a fraction, the denominator is the root of the number or letter, and then increase the response to the power of the numerator. Here is an example of a term written as an index: And finally, here is a problem where we have a number and a fraction index, four times X to half X half. So we leave the only four, the base is the same, so we can add the clues, half plus half which is one, so we have four Xs to one. Note that another important thing here is that X to the power of one is exactly the same as just X, so we can leave our answer as four Xs. so (1=2^{0}). In general, any expression with a null index is equal to 1. Also note that (0^{0}) is ambiguous and therefore we do not allow (a=0) in this law. Solution: [begin{align*} left(frac{3a^{3}b}{c^{2}}right)^{2}divleft(frac{ab}{3c^{-2}}right)^{-3} & =frac{3^{2}a^{6}b^{2}}{c^{4}}divfrac{a^{-3}b^{-3}}{-3}}{-3}}{3^{3}}} {3^{3}{^}{3^ {-3}C^{6}}QuadTextRM{by Law 3} & =Frac{3^{2}A^{6}B^{2}}{C^{4}}TimesFrac{3^{-3}C^{6}}{A^{-3}B^{-3}}QuadTextRM{Invertating the last term and Multiply} & =Frac{3^{ 2-3}A^{ 6}B^{2}C^{6}}{C^{4}A^{-3}B^{-3}}QuadTextrm{By Law 1} & =3^{-1}A^{6-left(-3right)}B^{2-left(-3right)}C^{6-4}quadtextrm{by law 2} & =3^{-1}A^{9}B^{5}C^{2}QuadTextrm{Simplifying} & =Frac{A^{9}B^{5}C^{2}}{3}quadtextrm{by negative index law} end{align*}] This formula tells us, that if a power of one number is increased to another power, multiply the indices. This is the fourth index law and is called the index law for powers.
(n) is the index in (a^{n}) with (a), which is called the database.