 Hello and welcome to the session. In this session we will discuss the powers and roots of signed numbers. First of all we shall discuss signed numbers. Positive and negative numbers are called signed numbers. A positive number is indicated either with no sign or with a plus sign preceding the number. For example, plus 4 or 4 is a positive number that is 4 units greater than 0 and a negative number is indicated with a minus sign preceding the number. For example, minus 4 is a negative number that is 4 units less than 0. Now we are going to discuss powers of signed numbers that is positive and negative numbers. Let us take any number say x and let us raise it to the power n. So x raised to the power n means we multiply x by itself n times that is x raised to the power n will be equal to x into x into x into x into x and so on into x n times. Let us take x to be a positive number say 3 then 3 raised to the power 2 will be equal to 3 into 3 that is equal to 9. Here we have multiplied 3 by itself. 3 raised to power 2 is called the square of 3. Similarly we have 3 raised to power 3 that is equal to 3 into 3 into 3 which is equal to 27. Here we have multiplied 3 by itself and then again by itself. 3 raised to power 3 is called the cube of 3. Similarly 3 raised to power 4 is called fourth power of 3. Thus 3 raised to power n is called n is power of 3. Now we will see effect of even and odd powers on signed numbers. Now let us take power n that is equal to 1, 2, 3, 4, 5, 6, 7 and so on. Now here 2, 4, 6 will be the even powers and 1, 3, 5, 7 will be the odd powers. Now let x be a positive number say 2. So 2 raised to the power 1 will be equal to 2, 2 raised to the power 2 will be equal to 2 into 2 that is equal to 4. 2 raised to the power 3 will be equal to 2 into 2 into 2 that is equal to 8. Similarly 2 raised to the power 4 is equal to 16. 2 raised to the power 5 is equal to 32. 2 raised to the power 6 is equal to 64. So we observe that when we raise a positive number to an even power or an odd power we obtain positive numbers only. See here the obtained numbers are 2, 4, 8, 16, 32, 64 are all positive. Thus a positive number raised to an even power or odd power gives positive number only. Now let x be a negative number say minus 2. So now minus 2 whole raised to the power 1 is equal to minus 2. Minus 2 whole raised to the power 2 will be equal to minus 2 into minus 2 that is equal to 4. Now minus 2 whole raised to the power 3 will be equal to minus 2 into minus 2 into minus 2 that is equal to minus 8. Similarly minus 2 whole raised to the power 4 is equal to 16. Minus 2 whole raised to the power 5 is equal to minus 32. Minus 2 whole raised to the power 6 is equal to 64. So we observe that when we raise a negative number to an even power we obtain positive numbers only. See minus 2 whole square is 4. Minus 2 whole raised to the power 4 is 16 and minus 2 whole raised to the power 6 is 64. Also when we raise a negative number to an odd power we obtain negative numbers only. See minus 2 whole raised to the power 1 is minus 2. Minus 2 whole raised to the power 3 is minus 8 and minus 2 whole raised to the power 5 is minus 32. Thus we conclude that a negative number raised to an even power gives positive number only and a negative number raised to an odd power gives negative number only. We must note that x raised to the power minus n can be written as 1 upon x raised to the power n. For example if we have 5 raised to the power minus 2 then it can be written as 1 upon 5 raised to the power 2 that is equal to 1 upon 25. Now we are going to discuss roots of science numbers that is positive and negative numbers. First of all we shall discuss square root. The inverse of squaring is finding the square root. We know that square of a positive or negative number is always positive. For example 4 square is equal to 16 and minus 4 square is also equal to 16. So here square roots of 16 are plus 4 and minus 4. Now plus 4 is the positive square root and minus 4 is the negative square root and we write square root of 16 which is denoted by this and this is equal to plus minus 4 or we can write it as 4 and minus 4. Thus 2 square roots of any positive number say n are given by square root of n and minus of square root of n where square root of n is positive root of n and minus square root of n is negative root of n. We must note that we cannot find the square root of a negative number because we cannot find any number say x whose square is negative. For example can we find x such that x square is equal to minus 4. Now if we take x is equal to 2 then 2 square will be equal to 2 into 2 that is equal to 4 and if we take x is equal to minus 2 then minus 2 whole square will be equal to minus 2 into minus 2 that is again equal to 4. In both cases we obtain positive 4 so we cannot find such x thus square root of any negative number does not exist. Now we are going to discuss cube root the inverse of cubing is finding the cube root. For example we know that 2 cube is equal to 8 thus cube root of 8 which is denoted by this will be equal to 2. Let us see cube root of a negative number we know that minus 2 whole cube is equal to minus 8 thus cube root of minus 8 is equal to minus 2. So we can say that cube root of a positive number is positive and cube root of a negative number is negative. Thus in this session we have discussed powers and roots of signs numbers. This completes our session hope you enjoyed this session.