Simplify two to the negative eighth power all divided by two to the negative third power plus two to the negative fourth power.
A number raised to a negative power is equal to the reciprocal of that number raised to the positive value of the power. So let’s begin with the numerator, two to the negative eighth power. So we need to take two and turn it into its reciprocal. Two is the same as two over one. And a reciprocal is where we will flip the fraction. So the reciprocal of two is one-half. So we take the reciprocal of that number and raise it to the positive value of the power. So we will raise this to a positive eight.
Now let’s do the same for the denominator. We have two to the negative third power. So the reciprocal of two is one-half. And the positive value of the power will be positive three. And then for two to the negative fourth power, the reciprocal will be one-half. And we’ll raise it to the positive four power.
Now we may be thinking there’re some properties that we could use where we could add the exponents or maybe subtract the exponents. But that’s what’s multiplying and dividing. And on the denominator, we’re adding with like bases. So this is different. So we need to simplify each of these and then try to put them together.
So one-half to the eighth power means we need to raise the one and the two each to the eighth power. And now we do the same for the numerator. We raise the one and the two to the third power and the one and the two to the fourth power. So we have one over 256 all divided by one-eighth plus one sixteenth.
So we need to add the fractions on the denominator the one-eighth and the one sixteenth. But in order to add fractions, we need to have a common denominator. So what is the smallest number that eight and 16 both go into? 16. So we don’t need to change the second fraction. It already has a denominator of 16.
But to change one-eighth to be something over 16, to get from eight to 16, we multiply by two, which means we need to do the same for the numerator. So for the denominator, we have two sixteenths plus one sixteenth. So we add the numerators of the two and the one to get three. And then we keep our common denominator of 16.
Now here we are dividing two fractions. Let’s go ahead and rewrite it this way, because when we divide by a fraction, we actually multiply by the second fraction’s reciprocal. So instead of division, let’s change it to multiplication. And we multiply by the second fraction’s reciprocal. So we need to flip the second fraction. So instead of three sixteenths, we’re multiplying by sixteen-thirds.
Now we could multiply straight across. But the 16 and the 256 can simplify. 16 goes into itself once. And 16 goes into 256 16 times. So on the numerators, we have one times one, which is one. And on the denominator, we have 16 times three, which is 48. Therefore, one forty-eighth will be our final answer.