Hello friends! Today we will learn the very first topic of Grade VII that is included in algebra. Algebra is a vast subject and students of this grade learn several topics in algebra that includes Problems Involving Positive Powers/Negative Powers, Solving problems in percent; proportional relationships, Formulas for measurement, Graphing data to demonstrate relationships, Arithmetic sequences, Simplifying numerical expressions, Operations on monomials, Linear/non-linear functions, equations, inequalities and Multi step problems. In this article we will discuss about problems involving negative and positive powers. Whenever we have terms raised to a number generally, we call them as exponents. The power raised to any number if contains negative sign then it is called as negative power problem and if the power raise to any number is positive then it is defined as the positive power problems.
Let's start the topic, you always write, 20 = 1 do you have any reason why we define 20 as 1 and what does a negative exponent mean? Whenever any number contains zero in its power it means that its value is one. Like 20 , 30 , 40, and so on will give answer as one only. You all know that exponents represent the numbers of times you have to multiply a number by itself. For illustration let's see an example: 32 this term means 3 raised to power 2. to simply the term we can write it as: 3 X 3 which will give result as: 9 similarly if we have 25 = 2X2X2X2X2. The positive power of any number represents the exact number of multiplication that a particular number have to go. This is the case when we have positive powers but what about negative powers? Like if you have 3-2, then how you will multiply three with negative two times.
A negative exponent is equivalent to the inverse of the same number with a positive exponent. For example: x-7, then it can be written as:
x-7 = 1/x7
You can easily solve these problems there is nothing special in case of negative power problems. It is a simple intermediate step which you may use. The best way to solve negative power problem is to work out lots of examples that will help you in solving lots of problems directly.
X-n = 1/Xn and 1/X-n = Xn,
Whenever you solve any problems related to powers be careful with negative power problems. The temptation is to negate the base, which would not be a correct thing to do. As you all know exponents are the other way to write, multiplication and the negative is in the exponent, to write it as a positive exponent we do the multiplicative
inverse which is to take the reciprocal of the base term.
Here, are few example of the positive power, like: 25 = 32, 24= 16,
and tell what is 23= ? , 22 = ? , 21 = ? and 20 =?. to solve such type of problems simply multiply the terms up to total number of powers,
like: 23 = 2 X2X2 = 4 X2 = 8, similarly, 22 = 2X2= 4, 21 = 2. And any number having power as zero we give answer as one always. Let’s take another example of this, 44 this can be written as 4 X 4X 4 X4= 16 X16 = 256. Whatever power is given to you simply multiply the number that much of times. When you do this you will easily solve the problems related to positive powers. In the same way we can have other such types of problems such as: 35 = 243, in this we multiply 3 five times. In this we will do 3X3X3X3X3= 243, as 9X3X3X3= 9X9X3= 81X3 = 243. In this way we can easily solve the problems related to exponents having positive powers. Students can try negative numbers i.e. negative base. You just divide by that negative number at each
step. For example: -25 = -32, similarly, -24 = 16, -23= -8, -22 = 4. In case of positive powers but negative base
we multiply the numbers as in case of positive powers only but, in this case you have to see whether the power is positive or negative. As you all know when we multiply to negative numbers they give positive result, this is defined in the rules of the mathematics. The other rule defines that if we have odd number then it gives the
negative output. Apart from this, you all must have read few other rules like:
-* - = +,
-* + = -,
+ * + = +,
+ * - = -.
These are the four main rules that are used throughout your mathematics. Whenever you solve any problem of mathematics the above rule plays a very vital role. So, friends whenever you multiply any number having such symbols, try to solve them following the rules. In case you multiply any number and calculate the answer correct and put the sign wrong then your answer is considered as wrong. Like: 2 * 3 = 6,
2 * -3 = -6,
-2 * 3 = -6, and -2 *-3 = 6
In this we see how an answer changes with the change of symbols. The simple multiplication can result different answers when its symbol changes.
Now, after this again move to our main topic of today. Negative powers, in this the power is negative base may or may not be negative as it depends on the problems. As I mentioned earlier that if we have X-n =
1/Xn, whenever we have negative power we take the term in the denominator and numerator is considered as 1in the above example. Let’s take few examples and understand the concept of negative powers properly. Suppose we have: 10-2
In this case we take 10 to denominator and then solve the problem,
1/102, Now, multiply the 10 with 10 this will give us: 1/10*10 = 1/100. In the same way you can solve many other problems like 2-2. In this you can easily multiply the 2 with 2 by taking it to the denominator as
½ * 2 = ¼, In the same way we can multiply other negative power sums by taking the negative term to the denominator and solve several math problems in seconds. Suppose if we have negative base and negative
power then how you will solve such type of problems. To understand this problem let’s take an example: (-3)-2 = 1 / (-3)2 = 1/9. In this we follow the same rule of negative power as I explained earlier and the concept or we can say rule of the operators. Example: (- 4)-3 = 1 /(- 4)3
In this we have power 3 so, 1 / -4 * -4 * -4 = 1/ -64.
As the power is odd so we get the sign of minus and if power is even we will get positive sign when base is negative.
Now, focus on few more example: simplify: 1 /(3) -3
In this question we are having negative power in denominator in this case we will solve the problem as first taking the denominator and convert it in numerator so that power will become positive, as 33
now simply multiply the positive powers as 3 * 3 *3 = 27.
Suppose we have z3 . z-5
This means we have z (3) + (-5)
= z-2
= 1 / z-2
Let’s have a complex example of this power problem: (x4 y3/ 2z2)-1 in this we have raise each base to -1.
(x4 y3/ 2z2)-1
= x(4 * -1) y (3 * -1)/ 2 -1 z(2 * -1),
= x-4 y-3 / 2-1 c-2 = 2c2 / a4 b3.
In this way students you can easily solve the several math problems having powers. Practice such type of problems and become master of mathematics. Negative powers and positive powers are used throughout mathematics in solving thousands of different problems. You can use this in solving differentiation and integration problems also. I tried my best to clear your doubts about this topic; still you have any problems then clear all your doubts and understand the concept properly so that you will not face any problem in future.
Students can take help of online tutors in order to learn different mathematics topics and in order to solve the several problems students can switch to online solvers and other tools. Friends do regular practice of different topics and become the master of mathematics. Practice is the main key to success, if you want to solve all math
problems and to score good grades do regular practice and feel how interesting is mathematics and how simple it is. In the coming articles we will discuss about, other algebraic, geometric, probability, statistics topics and I will make mathematics very easy for you. So practice negative and positive powers till we come with new problems
and topics. Enjoy mathematics!!
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