Explanation for Question 20 From the Math (No Calc) Section on the 2019 May Sat
Question 20 says if root X to the fifth divided by Q brew 2 X to the fourth is equal to X, 3 to the, a divided by B for all positive values of X. 4 What is the value of a over B? So this question requires you to 5 know a couple of exponent rules. So the first rule we're going to talk 6 about is changing the notation of a number to a power. 7 So roots X to the fifth can be rewritten 8 as X to the five over two. 9 And that's because you can always rewrite exponents or powers as fractions 10 where the numerator comes from the power under the radical. 11 And the denominator comes from the power of the root. 12 So X to the fifth to the cue or the square root becomes X 13 to the five over two. And we know that that's over cube root X 14 to the fourth, which just becomes X to the four over three, 15 based on that same rule. 16 The next rule you need to know is that when you have a number 17 to a PA or a two basis to two numbers that are the same 18 base, two different powers, and one is on top of the other, 19 their infraction form, then you can rewrite that using subtraction. 20 So this can become ax to the five over two minus four, 21 over three. And that's only because both the numerator and the denominator 22 have X as their base. So now that I have this rewritten as five 23 of the two minus four, over three, I can just take this top part 24 because I'm only looking for what a and B are equal to and simplify 25 it. So I have five over two minus four, 26 over three, I of course, need to change what both sides are written as 27 in order to subtract. So I'm going to multiply the left side by three, 28 over three in the right side by two, over two. 29 And that gives me 15 over six minus eight, 30 over six. And my final answer is seven over six. 31 So this can be rewritten as X to the seven, 32 over six, and therefore a over B is equal to seven 33 over six. And that's your final answer.