In the last line, it was easier to calculate 3 × 10 = 30 + 1 than to use false fractions. Once the null index and negative indices are introduced into algebra, all this can be replaced by the single law Algebraic fractions have been introduced in this module, but the algebra of algebraic fractions can cause considerable difficulties if it is introduced at full power before more basic abilities have been assimilated. Algebraic fractions are therefore examined in more detail in the fourth module, Special Extensions and Algebraic Fractions. Now that we are working in rational numbers – the integers with all positive and negative fractions – we finally have a system that is closed under the four operations of addition, subtraction, multiplication and division (except by zero). They are also concluded as part of the operation to support full digital powers. To calculate with indices, we need to be able to use the laws of indices in different ways. Let`s look at the different ways we can calculate with indices. However, fractions had not yet been introduced at that time, so the divisor could not be a power greater than the dividend. Now that we have fractions, we can express the law in broken notation and remove the limitation: fractional indices are powers of a term that are fractions.
Both parts of fracture performance have meaning. First, we need to make the index positive by writing the opposite. If the index is negative, set it above 1 and turn it over (write its reciprocal) to make it positive. Square roots and dice roots can also be represented with fractional indices – for example, is written as 5 and is written as 5 – and yet all five index laws apply even if the index is a rational number. Negative and fractional indexes are introduced in the Index and logarithms module. Later in the calculation, an index can be any real number. Take a look at the following expressions, which include simple fractional indices, and apply the laws of indices to simplify them as much as possible: The remaining index law of arithmetic deals with the powers of division of the same basis. The module multiple, factors and powers developed the law in the form “To divide powers of the same base, subtract the indices”. For example, the laws of indices are considerably extended as soon as negative and fractional indices are introduced into the Index module.
One clumsiness in this module concerned the power difference quotient, where we had to write three different examples. At the end, pay attention to the laws of indexes, spreadsheets, and exam questions. This module extends algebra methods so that all negative and decimal fractions can also be substituted in algebraic expressions and appear as solutions of algebraic equations. Here is an example of a term written as an index: The previous algebra module Negatives and index laws in algebra dealt with index laws for a power product of the same basis, for a power of a power and for a power of a product: There are two ways to construct rational numbers. The first is to build the positive breaks first, then the negative ones, which is pretty much what has happened historically. This involves a three-step procedure: An example of a fraction index is (g^{frac{1}{3}}). The denominator of the fraction is the root of the number or letter, and the numerator of the fraction is the power to increase the response. There are several laws of indices (sometimes called index rules), including multiplication, division, power of 0, parentheses, negative and fractional powers. In later modules on topics such as surds, circles, trigonometry, and logarithms, we need to introduce more numbers called irrational numbers and cannot be written as fractions.
Some examples of such numbers are An equation with several fractions can be solved by the standard method: “Move each term of x to one side and all constants to the other.” However, the resulting fracture stones can be quite complex. There is a much faster approach that eliminates all fractions in one step, a positive integer power tells you how much of the base you multiply together. To increase a fraction to a power, increase both the numerator and denominator to that power.1. Law of indices: (if the terms are multiplied, add up the powers)2. Law of Indices: (if the terms are divided, then subtract the powers)3. Law of indices: (if it is a power of a power, then multiply the powers)You can only apply the laws of indices if the bases are equal. Everything that is at the power of zero is 1. A negative power means a reciprocal: if it is an integer, then put 1 above; If it`s a break, flip it over (which amounts to the same). A broken power means a root; The root is the bottom bit, so the denominator tells you which root you need (square, cube, 4th, etc.). You also need to increase the value of the meter power – but of course, this only affects if the meter is not 1. We saw how to multiply fractions in arithmetic, we then constructed integer extensions in two different directions. First, in the fractions module, we added positive fractions like and = 4, which can be written as the ratio of an integer to a nonzero integer, and we showed how to add, subtract, multiply and divide fractions.
The result was a numeral system consisting of zero and all positive fractions. In this system, any non-zero number has a reciprocal, and division without remainder is always possible (wait by zero). However, only zero has an opposite and subtraction a − b is only possible if b ≤. Then we added the positive fractions to the integers so that our number system now contained all non-negative rational numbers, including numbers like 4 and 5. This system is closed under division (except by zero), but is still not closed under subtraction. Once fractions have been introduced in algebra, the two index laws dealing with quotients can be given in a more systematic form and then integrated into algebra. This concludes the discussion of index laws in arithmetic and algebra with respect to nonzero integer indices. The number of indexes tells us to find the square root, so although previous modules sometimes used negative fractions, this module provides the first systematic representation of these fractions and begins by representing the four arithmetic operations and powers in the context of negative fractions and decimals. The resulting numeral system is called rational numbers.
This system is sufficient for all normal calculations of all life, because adding, subtracting, multiplying and dividing rational numbers (apart from dividing by zero) and assuming integer powers of rational numbers always produces another rational number. In particular, scientific calculations often involve manipulating equations to the decimal place. Reciprocal positive and negative fractures are important in various subsequent applications, especially vertical gradients. When adding and subtracting fractions and decimals, which can be positive or negative, rules are a combination of rules for integers and rules for adding and subtracting positive fractions and decimals: when we look at the number of indexes, the denominator tells us to roll the root, and the numerator tells us, that we should square, so a quantity consisting of symbols with operations () is called an algebraic expression. We use the laws of indexes to simplify expressions with indexes. If fractions have no common denominator, we must find the lowest common denominator, just as we did for arithmetic fractions. We will limit the discussion here to numerical denominators and leave the denominators with numbers in the Special extensions and algebraic fractions module. Reciprocal algebraic fractions are formed in the same way as in arithmetic. All these numbers together are called rational numbers – the word “rational” is the adjective of “ratio”. We`ve actually been using negative fractions in these notes for quite some time.
These remarks are intended to formalize the situation. The usual sign laws apply to multiplication and division in this larger system, so that a fraction with negative integers in the numerator or denominator or in both can be written as a positive or negative fraction, arithmetic must be done in a system without such constraints – we want subtraction and division to be possible at any time (except division by zero). To achieve this, let`s start with the whole numbers and the positive breaks we already have. Then we add the opposites of the positive fractions so that our numeral system now contains all the numbers, such as: The module also begins the discussion about algebraic fractions, which regularly cause problems for students. In this module, the discussion is quickly limited by the exclusion of numerators and denominators with more than one term. These more complicated algebraic fractions are then discussed in more detail in the fourth module, Special Extensions and Algebraic Fractions. Apart from this qualification, substitution in algebraic fractions works in the same way as substitution in any algebraic expression.