True or false: If negative 10 cubed equals negative 1000, then the cube root of negative 1000 is equal to negative 10.
In this question, we’re considering 𝑛th roots. Here, we’re looking at the third root of negative 1000. Let’s recall how we define an 𝑛th root. An 𝑛th root 𝑟 of a quantity 𝑧 is a value such that 𝑧 is equal to 𝑟 to the power of 𝑛. Therefore, it’s the inverse operation to raising a number to the 𝑛th exponent. The 𝑛th root is denoted 𝑟 equals the 𝑛th root of 𝑧.
So, let’s consider what the values of 𝑧, 𝑟, and 𝑛 would be in the context of this problem. We can consider this general statement that 𝑧 is equal to 𝑟 to the power of 𝑛. Let’s compare it to the first statement: negative 10 cubed equals negative 1000. That means that 𝑟 would be equal to negative 10, 𝑛 would be equal to three, and 𝑧 would be equal to negative 1000.
We could then use this general statement that 𝑟 is equal to the 𝑛th root of 𝑧 in the context of this problem. Filling in the values then, we know that 𝑟 is equal to negative 10, 𝑛 is equal to three, and 𝑧 is equal to negative 1000. And this is equivalent to what we have in the second statement in the question problem. And so we can say that the statement must be true.
But we could also consider this in terms of the actual values. Let’s take the first statement: negative 10 cubed is equal to negative 1000. We can verify that this is true since negative 10 times negative 10 times negative 10 does indeed give us negative 1000. In order to find the cube root of negative 1000, we’re looking for a value which when written three times and multiplied would give negative 1000. The only value here which would work would be negative 10. So the cube root of negative 1000 is negative 10.
And so either by applying the 𝑛th root definition or by working out the values, we can demonstrate that this statement is true.