Because division is equal to multiplication by the reciprocal. A multiplication or division of the numerator of a fraction affect the fraction as a whole (and vice versa). Apply the commutative rule to  p − q. A division above equals a multiplication below. The order in which we write terms will not affect the sum. This problem illustrates the following theorem: In any equation we may change the signs on both sides. That is expressed in the following two rules. We say that −4 is the additive inverse of 4. We will have occasion to apply this theorem when we come to solve equations. We have seen the following rule for 0 (Lesson 3 ): 0 added to any number  does not change the number. Basic Algebra - A Simple Introduction to Algebra starting from simple arithmetic. (1 times any number does not change it. So, if we reverse a subtraction in both the numerator and denominator, the value of the fraction is unchanged. But instead of saying "obviously x=6", use this neat step-by-step approach: Work … Examples with step by step solutions, basic algebra review and worksheets. The rule of symmetry. Two fractions with common denominators can be added by adding the numerators and leaving the denominator unchanged. And so the rules of algebra tell us what we are allowed to write. Obviously, this will reduce the combined value of the exponent (for example, ``2^{4-2} = 2^2``). Happy calculating! (Lesson 2. Basic algebra – WJEC Algebra is very useful in the modern world where mathematics is used extensively. These basic rules are useful for everything from figuring out your gas mileage to acing your next math test — or even solving equations from the far reaches of theoretical physics. Therefore, any rule for addition is also a rule for subtraction. Since division is the inverse of multiplication, multiplying a number by itself a few times and then dividing it by itself multiplied a few time is the same as just multiplying it by itself a few less times. It will apply to any number of terms. Positivity is such a nice thing! The inverse of adding. If ``x = \sqrt{a}`` and ``y = \sqrt{b}`` then ``\sqrt{ab} = \sqrt{x^2*y^2}`` If we write out the multiplication, this turns into ``\sqrt{x*x*y*y}``. x is a variable. This is the algebraic version of the axiom of arithmetic and geometry: If equals are added to equals, the sums are equal. a + b − c + d  =  b + d + a − c  =  −c + a + d + b. To make things simple, we'll start with given values of ``m`` and ``n``. The commutative rule for addition is stated for the operation + . Thanks to the commutative property of multiplication, we can rearrange the Xs and Ys and get ``\sqrt{x*y*x*y} = \sqrt{(x*y)(x*y)} = \sqrt{(x*y)^2} = x*y = \sqrt{a}*\sqrt{b}``, Once again, by working backwards from the value of these two expressions we can see why they are equal. Nevertheless, it fits with the all-important exponent rule ``a^na^m = a^{n+m}``. Why may we subtract? If you were told that +4=10, you can probably see straight away that =6. Division is the inverse of multiplication: if ``{a \over b} = c`` then ``b*c = a``. And happily, ``\sqrt[mn]{x^{mn}} = x`` by definition, so we have ``\sqrt[m]{\sqrt[n]{a}} = x = \sqrt[mn]{x^{mn}}``. If x takes a positive value, then −x will be negative. We may multiply both sides of an equation by the same number. A LGEBRA, we can say, is a body of formal rules.They are rules that show … Example 1. ALGEBRA. And note that the symmetric version is also a rule of algebra. In the same fashion as the above rule, dividing the denominator of a fraction has the same effect as multiplying the numerator. All the most useful Rules of Algebra in one place: easy to understand, free, and accompanied by informative descriptions & examples. A fraction below equals a multiplication above. This becomes clear looking at the ``a^{n+m}`` side of the equation from rule 11. However, if we take a closer look at the rule ``a^na^m = a^{n+m}`` we can see that it implies that ``a^{-n}`` must equal ``{1 \over a^n}``, the multiplicative inverse or reciprocal of ``a^n``. SOME RULES. Problem 1. 0 is therefore called the identity of addition. Write the line that results from multiplying both sides by −1. Basic Algebra Algebra is about using letters in place of numbers. If you need to multiply a fraction, multiplying the numerator does the job. This means that we can take a multiplication raised to a power and rearrange the resulting series of multiplications to make two exponents, It might seem odd to have a negative exponent (since you can't multiply something by itself a negative number of times).