Question Video: Evaluating Cubic Roots of Algebraic Expressions Mathematics

Video Transcript

Which of the following has the same value as the cube root of 729𝑚 to the power of 12 𝑛 to the power of 21? Option (A) the length of one side of a square whose area is 729𝑚 to the power of 12 𝑛 to the power of 21. Option (B) the length of one side of a cube whose volume is 729𝑚 to the power of 12 𝑛 to the power of 21. Option (C) the expression 243𝑚 to the power of four 𝑛 to the power of seven. Or option (D) the expression 729𝑚 to the power of 12 𝑛 to the power of 21.

Let’s begin by taking a look at this rather large expression, which has an 𝑛th root, more specifically a cube root. In fact, we can consider this expression to be the cube root of each of these terms within the expression. We can say this because of this exponent rule, which tells us that for any real values of the 𝑛th root of 𝑎 and the 𝑛th root of 𝑏, we could say that the 𝑛th root of 𝑎 multiplied by the 𝑛th root of 𝑏 is equal to the 𝑛th root of 𝑎𝑏. We can also apply this in the reverse direction. That is, we can say that the 𝑛th root of 𝑎𝑏 must be equal to the 𝑛th root of 𝑎 multiplied by the 𝑛th root of 𝑏.

In this problem, we can therefore say that the expression in the question is equal to the cube root of 729 times the cube root of 𝑚 to the power of 12 times the cube root of 𝑛 to the power of 21. We can then take each part on the right-hand side and evaluate it. 729 might not be one of the most common cube numbers that we know, but in fact the cube root of 729 is nine. Next, in order to evaluate the cube root of 𝑚 to the power of 12, we’ll need to use a few more exponent rules.

Let’s firstly recall that we can write the 𝑛th root of 𝑎 as 𝑎 to the power of one over 𝑛. And so the cube root of 𝑚 to the power of 12 is equivalent to 𝑚 to the power of 12 to the power of one-third. And in fact we can simplify this even further. Let’s use the rule that 𝑥 to the power of 𝑎 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎𝑏. This means that we can multiply the exponents of 12 and one-third. 12 times one-third is four. And so the second term on the right-hand side can be simply written as 𝑚 to the power of four. We can perform the same process to simplify the cube root of 𝑛 to the power of 21. Writing the cube root as a fractional index, we have 𝑛 to the power of 21 to the power of one-third. Then, multiplying the exponents of 21 and one-third, we get seven.

We can then write nine times 𝑚 to the power of four times 𝑛 to the power of seven in a nicer way, as nine 𝑚 to the power of four 𝑛 to the power of seven. But we might notice that this answer doesn’t look really very similar to any of the answer options that we were given. However, we can definitely eliminate answer options (C) and (D) because neither of these expressions are equivalent to the expression that we have worked out. In fact, have you noticed the very common error in option (C)? The value of 243 comes from incorrectly finding one-third of 729 instead of taking its cube root. The incorrect expression in option (D) is simply the expression which was within the cube root in the question. These two expressions are not equivalent.

Let’s take a closer look at option (A), which says that the expression in the question is the length of one side of a square whose area is 729𝑚 to the power of 12 𝑛 to the power of 21. We should remember that if we have a square of side length 𝑙, then its area is equal to 𝑙 squared. That means that if we were given the area of a square and we wanted to work out its side length, we would take the square root of the area. That means that if we did have a square with this given area, then to find its side length, we would take the square root. And of course, in the question, we didn’t have the square root of this expression. We had its cube root. Therefore, we can eliminate answer option (A).

This leaves us with one remaining answer option. So let’s check that it is indeed the correct one. This option says that the expression we were given in the question is the length of one side of a cube whose volume is 729𝑚 to the power of 12 𝑛 to the power of 21. Let’s remind ourselves of how we would work out the volume of a cube of side length 𝑙. Because it’s a cube, we know that all sides will be the same length. Therefore, the volume is equal to 𝑙 cubed. Of course, if we were given the volume and we wanted to work out the side length, we would take the cube root of the volume.

And so if we did have a cube with a volume of 729𝑚 to the power of 12 𝑛 to the power of 21, then the length of one side would indeed be the cube root of 729𝑚 to the power of 12 𝑛 to the power of 21. We have therefore confirmed that the answer given in option (B) is the correct one.