IX grade Powers and Radicals

Friends today we all are going to learn one of the most common and important topic of IX grade mathematics which requires lot of practice and focus. Powers and Radicals are the two topics which we are going to discuss, as both topics come in almost each and every area of mathematics studies. Sometimes problem becomes so tough to solve as it might come with powers or in radical form. A daily practice and hard work will help you to get some confidence. Let's start with powers and will talk radicals later. It offers an admirable and compact explanation of two important principles in mathematics :

1. The first concept is to introduce concepts in a simple manner and then generalize them in such a way that rules and facts that are true in the basic manner remain true in the more general context.
2. The another concept is to reduce your problem to one you have already solved before.
   This is the way through which we can obtain powerful and efficient problem solving tecjniques and we can further improve our understanding of mathematics.

Now the topic to study is Multiplication : While doing multiplication we observe that it is defined by repeated addition. Thus for any number “x”

3 * x = x + x + x.

In a more pronounced manner, for any number “x” and any of the natural number n (that is one of 1,2,3,....) we define the product

n * x = x + x+ …..+x

here X is repeated n times.

Now we can extend this definition to factors n that are not natural numbers such as negative numbers, fractions, or, in general, real numbers etc. in such a manner that the ordinary rules of arithmetic always remain true.

The next topic to discuss is Natural Number Exponents

Powers work exactly the same manner except that instead of addition we use multiplication as our basic repeated operations. Thus, for any of the number x and any natural number n.

xⁿ = x * x * …...* x

Here n factors x

Lets take an example to understand:
2³ = 2 x 2 x 2 = 8
3² = 3 x 3 = 9
1ⁿ = 1 x 1 x …..x 1 = 1
0ⁿ = 0 x 0 x 0 x 0 = 0
x¹ = x
The expression Xⁿ is called a power and described in words as x to the power n. The number x is the base of the power and the number n is its exponent.

Now the next topic is the Central Rule :

2³ x 2⁴ = (2 x 2 x 2) x (2 x 2 x 2 x 2) = 2^{3 + 4} = 2⁷

For any real number x and natural numbers m and n

x^m x xⁿ = x^m+n

The above formula is called as central rule. We write down m factors x and then n factors x , and so we need to write down a total of m + n factors x.

Next topic is Zero Exponents

Now we are going to understand the case where the exponent is having a value zero. You should concentrate a bit more on this paragraph. It is elementary and easy to understand, and yet it shows a broad folder of mathematical thinking. Here we need to apply our central rule. Let's take an example :

2⁰ x 2³ = 2⁰⁺³ = 2³

 Here 2⁰ is basically a number that gives 2³ when multiplied with 2³. There is only one number that gives namely the number 1. So we are compelled to define 2⁰ = 1 . If a base was a number other than 2, say x, the same sort of argument applies, and so we define z⁰ = 1. There is one caution, however. If this x = 0 definition would give 0⁰ = 1. We also see that for any natural number n. we have that 0ⁿ = 0. If we were to expand that rule to n = 0 we would have 0ⁿ = 0. So we obtain different value of 0⁰, depending on which rule we want to apply. In this case, neither rule is better than the other, and so the most useful approach is to resign ourselves to the fact that 0⁰ is undefined. There is no problem when the base is non-zero however, and so we define

x⁰ = 1 (for all x not equal to 0)

Now I am going to teach the basic concept that is the exponent of a number says how many times to use the number in a multiplication.

Let's take an example to understand it better : 3² can be written as 3 x 3 = 9. 3² could be called as "3 to the second power", "3 to the power 2" or simply "3 squared".

Y = e^x The variable x in the equation is called the 'exponent', the function is called the exponential function.

The inverse of the exponential function is the logarithmic function or logarithm. This general logarithm is written as y = ^a log(x) or y = log _a (x).

The logarithmic function is defined for all positive real numbers x. So, the domain of the logarithmic function is the set of all positive real numbers. Operations as multiplying, dividing or raising to a power can be simplified to the adding and subtracting of logarithms by use or relations as log ab = log a + log b and log a^k = k log a.

Now the next topic is Scientific Notations : Scientific notation is a method developed by scientist. Scientists have developed a shorter method to express very large numbers. A number in scientific notation is written as the product of a number (integer or decimal) and a power of 10. This number is always 1 or more and less than 10. Scientific Notation is based on power of the base number 10.

The number 23,000,000,000 in scientific notation is written as:

2. 3. x 1011

here the first number 2.3 is called the coefficient. It must be greater than or equal to 1 and less than 10.

The second number is called as the base value. It must always be 10 in scientific notation. The base number 10 is always written in exponent form. The number 11 is referred to as the exponent or power of ten.

Following rules are applied when Scientific notation problems come into account:
Addition and Subtraction: All numbers are converted to the same power of 10, and the digit terms are added or subtracted.
Multiplication: The digit terms are multiplied in the normal way and the exponents are added.
Division: The digit terms are divided in the normal way and the exponents are subtracted.
Power of Exponentials: The digit term is raised to the indicated power and the exponent is multiplied by the number that indicates the power.

Roots of calculations : We need to change the exponent if necessary so that the number is divisible by the root.

Next topic we will study is radicals:

First question comes in our mind is that what is a Radical? Radical is known as root of an expression. A fraction is not a radical, but a fraction may contain a radical. It uses the the sign of radical (√) also known as surds. To understand it better let's take an example:

squares 2 = 4 it means 2 is the square root of four. 2³ = 8 this means 2 is the cube root of 8. if a, b are real numbers, n is a positive integer and if an = b then the n th root of b is a. then it can be written in this form :

n√b = a

n√b the radical sign, n is the indexed and b is the radicand. The index give the degree of the roots.

A radical equation is an equation in which at least one variable expression is stuck inside a radical, usually a square root. For example

√x + 2 = 6

2 + 2 = 4

( 2 + 2 )² = squared 4 here the final result shows that

16 = 16.

Various online radicals calculators are available which helps us to solve radicals. Simplifying Radical Calculator helps in getting an accurate answer in faster manner. It also provides the complete solution which helps us to solve radicals manually. By taking an Online Math help we can easily understand the basic concept behind the radicals equations or problems.

The steps to solve radical problems are: break or isolate the radicals to the left side of equal signs means get one radical on one side and everything else on the other using inverse operations. Then square each side of the equation and solve the equation we get all the solution.

Equation which contains radicals deals with square root equations whereas the square root symbol is known as radicals. Radicals also plays an important role in various engineering and physics branches. Various algebraic equations also come with the inclusion of radicals.