in today's video we're going to talk about negative exponents and reciprocals to start off let's talk about what a reciprocal actually is if I have the fraction two-thirds the reciprocal of two-thirds is three over two in other words it's two numbers whose product is equal to one and in another way of putting it we just make the numerator the denominator and the denominator the numerator it's as simple as that we're going to use uh this concept when talking about negative exponents um it's not good practice to leave an expression with any negative exponent so we're going to talk about how to how to change an expression so that it doesn't have any negative exponents anymore so if we're given an expression 7 to the exponent of negative 2 . we can represent anything as as a fraction so I'm going to just put this all over 1. and the rule we're going to talk about today is basically if we have an exponent a negative exponent we can take the reciprocal of the expression and then what happens is is the exponent changes sign so in other words it can go from negative to positive so in general x to the negative n will equal 1 over x to the positive n so in our example here this is going to equal 1 over 7 squared so we take the reciprocal of the expression and what's in the numerator goes to the denominator and vice versa and what happens is now our exponent instead of being negative 2 it's a positive 2. and now we can evaluate this it becomes 1 over 49. okay let's look at another example so if we have 10 over 3 all to the exponent of negative 3. we could apply that negative 3 to both the 10 and the 3 but before we do that we're going to apply what we just learned we want to eliminate that negative exponent so in order to eliminate the negative exponent we take the reciprocal of 10 over 3. and now that we've taken the reciprocal we can change the sign of the exponent right be careful here a common mistake is is for students to change the the sign of the numbers inside the brackets we we don't change the 10 over 3 right we're only changing the sign of the exponent so it becomes cubed instead of to the exponent of negative 3. now we can apply the cube to both the numerator and the denominator so we will get 3 cubed which is equal to 27. over 10 cubed which is equal to a thousand and that's our answer let's do a couple more negative 1.5 to the exponent of negative 4. so we can what we're going to do here first is we're going to convert the decimal into a fraction so negative 1.5 is the same thing as negative 3 over 2. it's rewriting that and now we can proceed with what we learned uh previously we want to change that exponent so that it's positive and not negative so we're going to take the reciprocal of our expression so we'll have 2 over negative 3. and change the sign of the exponent and again that negative can go wherever it wants it's we have negative it's negative 1.5 so as long as one of the values you know is negative that's fine so now we have we can apply that exponent to each of our our terms so the numerator and the denominator we will be left with 2 to the exponent 4 which is 16. and negative 3 to the exponent 4 which will equal positive 81. any even exponent applied to a negative number will result in a positive let's do one more 25 over 36 all to the exponent of negative one-half so the first step we want to do is we're going to take the reciprocal by taking the reciprocal the exponent now becomes positive and now we can apply that exponent to the numerator and the denominator we learned in a previous lesson that any number to the one-half is the same thing as the square root at the square root of 36 over the square root of 25 which is equal to 6 over 5. thank you