Simplify the cube root of negative 0.064𝑎 to the power of 30.
In order to try and solve this problem, we’re actually gonna look at splitting into two parts, just so you can see how, first of all, you’d find the cube root of negative 0.064. And then, how you’d find the cube root of 𝑎 to the power of 30. And then we’ll bring it back together to actually give the final answer.
We start on the left-hand side. And first of all, if we’re gonna try and find the cube root of this number, we’ll look at the 64 and we’d say, “well actually, 64, we know that the number of this cube root of 64 is four.” So four is gonna have to be involved. Also we can see that there are three decimal places. Well, this must mean that our answer has one decimal place because it’s gonna be multiplied by itself three times. So that again, would give us our three decimal places, so we get 0.4. And the final bit of information we have to consider is the negative sign. And therefore, we know that our answer must be negative, as a negative multiplied by a negative multiplied by a negative gives us a negative. And that’s because a negative multiplied by negative is positive, then multiplied by a negative again give us our negative answer. Great, so we’ve now found negative 0.4 to be the answer for our left-hand side.
Now we’ll look at the right-hand side and look at the cube root of 𝑎 to the power 30. Okay. To actually work this out, we’re gonna use this rule which says that the 𝑎 root of 𝑥 is equal to 𝑥 to the power of one over 𝑎 which would then mean that our cube root of 𝑎 to the power 30 is gonna be equal to 𝑎 to the power 30 but to the power of a third, and remembering the parentheses there.
Next, we’re gonna use this rule which shows us that 𝑥 to the power of 𝑎 inside parenthesis to the power of 𝑏, which is outside the parenthesis, is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. So we can apply this rule to what we’ve got here. So this will give us 𝑎 to the power of 30 multiplied by a third, which is gonna give us 𝑎 to the power of 10 because a third of 30 is 10. So then we can put it together to get our final answer.
So we can say that the cube root of negative 0.064𝑎 to the power of 30 is equal to negative 0.4𝑎 to the power of 10.