True or False: The fifth root of 243 plus the fourth root of 81 subtract the square root of nine is equal to the cube root of negative 27.
In this question, we’re given four different 𝑛th roots. We can recall that the 𝑛th root 𝑟 of a quantity 𝑧 is a value such that 𝑧 is equal to 𝑟 to the power of 𝑛. To find 𝑟, we calculate the 𝑛th root of 𝑧. So let’s begin this question by working out the value of each of these three expressions on the left-hand side.
Let’s say that the fifth root of 243 is equal to a root which we’ll define as 𝑎. Therefore, 𝑎 to the power of five is equal to 243. And we just need to work out what 𝑎 would be. When we take a value to the power of five, that means that we have that value five times multiplied together. When we work a lot with powers, we begin to memorize what some common powers are and their results. However, if we don’t know, then it’s a good idea to start with the trial and error method.
We can discount one to the power of five since one times one times one times one times one would give us one. Then we might realize that it can’t be a power of two because any power of two will be an even number and 243 is odd. So what if the value of 𝑎 was three? Well, if we take the first three times three, that would give us nine. Then when we multiply that by another three, we’ll get 27. Multiplying by the fourth three, we’d get 81. Finally, multiplying 81 by three gives us 243. Therefore, we’ve worked out the value of 𝑎 must be three since it’s three to the power of five that equals 243. Therefore, we can say that the fifth root of 243 must be three.
Next, we could have a look at calculating the fourth root of 81. This time, let’s say that that fourth root must be equal to 𝑏. We’re therefore looking for some value 𝑏 to the fourth power which gives us 81. You might remember that we have actually just seen the value of 81 in terms of a fourth power. In our earlier calculation, we worked out that three times three times three times three gives us 81. So three to the power of four is equal to 81. And so this fourth root of 81 is also three.
Now we have one last term remaining on the left-hand side. It’s the square root of nine. When we write the square root of a number, we don’t have a smaller number within the root sign. We can, however, write a smaller two if we wish. So if the square root of nine is equal to a root 𝑐, then that means that we’re looking for a value 𝑐 squared which would be equal to nine. We should remember that three times three gives us nine. Since three squared is nine, then that means that the square root of nine is three.
As an aside, you might remember that there is another value squared which gives us nine, since negative three squared would equal nine. However, unless we’re told otherwise, we usually just consider the positive value of the root.
Now that we have the three values on the left-hand side, let’s consider what the value of this expression the cube root of negative 27 would be. This time, let’s say that the cube root of negative 27 is equal to some root 𝑑 such that 𝑑 cubed would give us negative 27. We might remember that we can’t take the square root of a negative number, but we can take a cube root of a negative number. In fact, we can take any root of a negative number so long as the root is an odd value.
We should remember that three cubed, that’s three times three times three, gives us positive 27. Therefore, negative three cubed would give us negative 27. And so the cube root of negative 27 is equal to negative three. Now we can substitute the four values that we have for each of these expressions. We must be careful when we’re writing out this calculation as it would be very easy not to notice that this second symbol would be a subtraction rather than an addition.
So on the left-hand side, three plus three is six subtract three gives us three. And on the right-hand side, we have negative three. However, the left-hand side is not equal to the right-hand side. Therefore, we can give the answer false since the given equation statement does not hold true.