Problem number 7 states, let x equal pi over 4. Which of the following numbers is the largest? So we have our x value, which is pi over 4. And pi is going to be 3.1415 and some other decimal points. So we know that pi is going to be less than 4. And if the denominator is less than the numerator, then we know that the number is less than 1. So now we know our x value, pi over 4, is less than 1. For values less than 1, if they are raised to some power, in this case we will do 2, we know that we will get a smaller number. So 0.5 to the 2nd power is going to be equal to 0.25, a half of a half. If we make this an even greater power, for example 0.5 to the 4th power, then the number is going to be even smaller, we'll get 0.0625. So we know that x to the 4th is going to be less than x squared, which is going to be less than just x. As a result, it cannot be x to the 4th or x squared because x is greater than them. Now let's try doing the roots. Taking the root of something is equivalent to raising it to the 1 half power, which is like asking which number times itself will equal 0.5. And since we know that numbers less than 1 when multiplied by themselves result in numbers smaller than themselves, we know this value must be greater than 0.5. The exact value is some decimal. So if we go 0.5 to the 1 4th power, we know that the resulting value will be much greater than 0.5, still less than 1 though. So when x is less than 1, which it is, we know the 4th root of x will result in the largest number. So the question asked us, which of the following numbers is the largest? The answer is the 4th root of x, letter e.

powers

roots

comparison

fourth root

largest value