 Hi and welcome to the session. Today we will learn about Exponents and Powers. Try to read this number. Don't you think this is difficult to read this number? So, to make this task easier and simpler, we will study Exponents and Powers. First of all, let us start with a small number, say 1 lakh. Now, we can write this as 10 into 10 into 10 into 10. Here we have multiplied 10 by itself 5 times. So, we can write it as 10 raised to the power 5. We can also read it as 5th power of 10. Now, here 10 raised to the power 5 is known as the exponential form of 1 lakh. Here we have 10 raised to the power 5 in this 10 is called the base and 5 is called the exponent. Now, suppose we want to express 16 in exponential form. Then, we can write 16 as 2 into 2 into 2 into 2. Here we have multiplied 2 by itself 4 times. So, this will be equal to 2 raised to the power 4, where 2 is the base and 4 is the exponent. Similarly, we can take a negative integer. Suppose we want to find out minus 3 raised to the power 3. This will be equal to minus 3 into minus 3 into minus 3, which will be equal to minus 27. Now, some powers are given special names. Suppose we have an integer a, then a raised to the power 2 will be read as a squared and a raised to the power 3 will be read as a cubed. Now, suppose we are given a number 3, 2, 5, 4, 7 and if we want to write it in expanded form, then this will be equal to 3 into 10,000 plus 2 into 1000 plus 5 into 100 plus 4 into 10 plus 7. Now, using exponential form, we can write it as 3 into 10 raised to the power 4 plus 2 into 10 raised to the power 3 plus 5 into 10 raised to the power 2 plus 4 into 10 plus 7. Next, there is an important point, which you need to remember. Suppose we have minus 1 as the base and the exponent of minus 1 is an odd number, then this will be equal to minus 1 and if the exponent of minus 1 is an even number, then this will be equal to plus 1, that is 1. For example, minus 1 raised to the power 51 will be equal to minus 1 because 51 is an odd number and minus 1 raised to the power 98 will be equal to plus 1, that is 1 because 98 is an even number. Our next topic is laws of exponents. So, first of all, let us see multiplying powers with the same ways for any nonzero integer a raised to the power m into a raised to the power n will be equal to a raised to the power m plus n where m and n are whole numbers. For example, 5 raised to the power 9 into 5 raised to the power 100 will be equal to 5 raised to the power 9 plus 100 which will be equal to 5 raised to the power 109. Next, we have dividing powers with the same ways. For any nonzero integer a raised to the power m divided by a raised to the power m will be equal to a raised to the power m minus n where m and n are whole numbers and m is greater than n. For example, minus 2 raised to the power 100 divided by minus 2 raised to the power 9 will be equal to minus 2 raised to the power 100 minus 9 which will be equal to minus 2 raised to the power 91. Next, we have taking power of a power. For any nonzero integer a raised to the power m whole raised to the power n will be equal to a raised to the power m into n where m and n are whole numbers. For example, 15 raised to the power 9 whole raised to the power 2 will be equal to 15 raised to the power 9 into 2 which will be equal to 15 raised to the power 18. Let's move on to multiplying powers with the same exponents. For any nonzero integers a and b raised to the power m into b raised to the power m is equal to a into b whole raised to the power m where m is any whole number. For example, 2 raised to the power 5 into 3 raised to the power 5 will be equal to 2 into 3 raised to the power 5 which will be equal to 6 raised to the power 5. Next is dividing powers with the same exponents. For any nonzero integers a and b raised to the power m divided by b raised to the power m is equal to a raised to the power m upon b raised to the power m which is equal to a upon b whole raised to the power m where m is any whole number. For example, 15 raised to the power 9 divided by 3 raised to the power 9 will be equal to 15 upon 3 whole raised to the power 9 and this will be equal to 5 raised to the power 9. Lastly we have numbers with exponent 0. For any nonzero integer a raised to the power 0 is equal to 1. So that means if we have any nonzero integer then that integer raised to the power 0 will always be equal to 1. For example, minus 352 raised to the power 0 will be equal to 1. Let's move on to our next topic, decimal number system. We have already learnt to write a number in expanded form using exponential form. That is the number 32547 can be written as 3 into 10 raised to the power 4 plus 2 into 10 raised to the power 3 plus 5 into 10 raised to the power 2 plus 4 into 10 plus 7. Now using the laws of exponents we can rewrite it as 3 into 10 raised to the power 4 plus 2 into 10 raised to the power 3 plus 5 into 10 raised to the power 2 plus 4 into 10 raised to the power 1 plus 7 into 10 raised to the power 0 because 10 raised to the power 0 is 1 and 7 into 1 is 7. So here we started from the maximum value of the exponent of 10 that is 4 and went up to 0 in the last step. Now let's see how to express last numbers in this standard form. Let us start with a small number say 32547. We can write this number as 3.2547 into 10000 that is 3.2547 into 10 raised to the power 4. So here this form is known as standard form. So to write a number in standard form we must express the number as a decimal number between 1 and 10 including 1 multiplied by a power of 10. Now let's see how to find the exponent of 10 while writing a given number in standard form. Suppose we want to express a number 325.47 in standard form. Here to find out the exponent of 10 first of all we will count the number of digits on the left of decimal point in the given number which is 3 over here. Then we will subtract 1 from this and we will get 2. So 2 is the exponent of 10 for the given number and thus this number will be equal to 3.2547 into 10 raised to the power 2 in the standard form. Now let's go back to this number. Here we can say that the decimal point is over here on the extreme right of the given number. That means there are 5 digits on the left of decimal point and thus the exponent of 10 will be 5 minus 1 equal to 4. So here 4 is the exponent of 10. Now let us express this given number in standard form. So first of all here we can see that the decimal point is over here on the extreme right of the number. Now we need to count the number of digits on the left of decimal point which is 15 over here. So that means the exponent of 10 will be 15 minus 1 that is 14. So this number will be written as 5.024 into 10 raised to the power 14 in the standard form. With this we finished this session. Hope you must have understood all the concepts. Goodbye, take care and have a nice day.