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.

Introduction to IX Grade Algebra

Friends today we all are going to rewind our basic concepts of Algebra along with ninth standard Algebra problems. We all know about the basic concepts behind numbers and how can we combine them with the help of basic operations of addition, multiplication, subtraction and division. This mathematics area of study is known as Arithmetic. In ninth standard we are going to study the more advanced area of Algebra which is distinct from arithmetic. In this section of Algebra we are going to perform basic operations for example addition to specific numbers that involves entities called as variables. Variables are those entities which have no particular value or an unknown value. The variables in mathematics are usually denoted by the upper or lower case letters. Algebra is a fundamental branch of mathematics that mainly deals with the rules of operations and their relations and the constructions and concepts arising from them. If we talk in simple words then we will see that algebra is simply the art of replacing variables in place of numbers.

Now I am going to discuss it in deep. An algebraic expression is basically a collection of numbers and letters combined by the four (addition, multiplication, subtraction and division) basic arithmetic operations. Here are some of the examples of algebraic expressions to understand it better.

7y, 3a+y, 3a-4y, a/(a+y), a², (a+b)²
The numbers used in the algebraic expression are called constants. Additionally, if there is no variable in the algebraic expression then we called it an arithmetic expression. For example 4 + 3 / 8.

Variables in the algebraic expressions are basically used to describe general conditions or situations or we can say that it models the real problems in the form of algebraic expressions and they can be used to solve problems that in other manner would be much more difficult or complicated or even impossible. These applications are used when we all are going to solve word problems.

Now moving forward let’s discuss about one of the most important and frequently used terminology that is an equation. An equation is basically an allegation that two algebraic expressions are equal or similar. This statement can be represented in two different ways or we can say that it can have two distinct meanings:

The first statement says that the equation is true for all values of the variables. In such kind of situations the equation is called an identity. An example of an identity to understand the above statement is
x + y = y + x
for all numbers x and y. This statement or example is called the commutative law of addition. Another well known identity is the first binomial formula which can be stated as :
 (a+b)² = a= + 2ab + b²
The second statement says that the equation is true for some of the values of the variables. Here we need to find out the values of the variables for which the equation is true. This terminology is called as solving the equation. Let’s take an example to understand it better. The equation to solve is: 3y + 1 = 4
in this equation we know that
y = 1
is the solution. The above equation is an example of a linear equation. Let’s take a bit more complicated example of a quadratic equation is y² - y - 2 = 0. Now after solving this we found that this equation has the two solutions
  y = -1    or   y = 2.
Now in Ninth standard we are also going to understand the basic concept behind evaluating an algebraic expression. To evaluate an algebraic expression can be stated as, substituting specific values for its variables. Lets’ take a very simple example to understand it:
Expression = 2y + 1
Evaluating expression, for the value of variable y = 3
give expression = 2 * 3 + 1= 7
We can say that value of expression at y = 3 is 7.
Let’s take an another way to solve which is a bit more complicated:
Expression at y = 2x + 1, where x is another variable
Expression = 2 * (2x + 1) + 1 = 4x + 2 + 1 = 4x + 3.

The next condition is if there are equivalent expressions:

Two expressions are equivalent if their values are equal for all possible estimations of the two expressions. In a simple mathematical manner we can say that listing expressions with an equality sign between them always gives an identity.

Now the next topic of the Algebra section is Number system. This section tells about the different types of numbers. Here we are going to talk about the four types of numbers that are Natural numbers, the Integers, rational numbers and real numbers.
Natural Numbers: 1, 2, 3, 4, 5, ..... are the natural numbers. We can add or multiply the two given natural numbers and obtain another natural number. Some of the mathematical laws to follow with natural numbers are:
x + y = y + x The commutative law of addition
x * y = y * x The commutative law of multiplication
(x + y) + z = x + (y + z) The associative law of addition
(x * y) * z = x * (y * z) The associative law of multiplication
(x + y) * z = x * y + y * z The distributive law
The Integers: It includes all the natural numbers, the zero number and all the negatives of the natural numbers. The example for this is : ..., -3, -2, -1, 0, 1, 2, 3, ....
Rational Numbers: The rational numbers can be defined as the ratios of integers. The most important condition is that denominator can never be a zero.
Real Numbers: It is basically a decimal expression whose digits or values may or may not terminate or repeat.




Let’s talk about scientific Notation: It is a way of writing numbers that accommodates or houses values too large or small to be conveniently written in standard decimal notation. Or in simple language a Scientific notation is used to express very large or very small 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 powers of the base number 10.
The number 33,000,000,000 in scientific notation is written as:
3.3 x 10¹¹
Here the first number 3.3 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.

Now the another section that is Logarithms:
The logarithm of a number is basically an exponent of a number or we can say that an exponent by which another fixed value, the base of the value, need to be raised to produce that number. In simple mathematical manner we can say that the logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. In mathematical view : b^y = x.

Now we are going to discuss about Logarithmic Functions in mathematical world. We can define logarithmic function as the inverse of the exponential function. In the above formula:
log_b(x) which is equivalent to x = b^y.
The logarithm to the base e (exponent) is written as l_n(x).

A brief introduction on Radicals:
Radical is known as root of an expression. A fraction is not a radical, but a fraction may contain a radical. It uses the sign of radical (√) also known as surds. Let's take an example to understand it better. A radical equation is an equation in which at least one variable expression is stuck inside a radical, usually a square root.
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 root b = a
In n root b, n is the indexed and b is the radicand. The index gives the degree of the roots.

For example : Root x + 2 = 4
root x = 4
root x = squared 2
x = 2.
This is all about basic ninth grade Algebra, moving further we all are going to learn equation section of ninth grade algebra.