Question Video: Rewriting a Radical in a Simplified Form Mathematics

Write βˆ›2 Γ— βˆ›16 in the form π‘Žβˆ›π‘, where π‘Ž and 𝑏 are two integers and 𝑏 is the smallest possible positive value.

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Video Transcript

Write the cube root of two times the cube root of 16 in the form π‘Ž times the cube root of 𝑏, where π‘Ž and 𝑏 are two integers and 𝑏 is the smallest possible positive value.

In this question, we’re given an expression involving the product of two cube roots. And we are asked to rewrite this expression in the form π‘Ž times the cube root of 𝑏, where π‘Ž and 𝑏 are integers and 𝑏 is the smallest possible positive integer where this is true.

To answer this question, we can start by noting that the form we need to rewrite the expression is often thought of as the standard or simplified form of the cube root of an integer. In general, to write an expression in this form, we first need to write it as the cube root of an integer. This means that we want to start by rewriting the given expression as the cube root of an integer.

To do this, we can recall that for any real numbers π‘₯ and 𝑦, the cube root of π‘₯ times the cube root of 𝑦 is equal to the cube root of π‘₯ times 𝑦. Applying this result with π‘₯ equal to two and 𝑦 equal to 16 gives us that the cube root of two times the cube root of 16 is equal to the cube root of two times 16, which we can simplify to give the cube root of 32. This is now in the form of the cube root of an integer.

We can rewrite the cube root of an integer by using our result for the product of cubes in reverse. Imagine that we are taking the cube root of an integer that we can rewrite in the form the cube root of 𝑐 cubed times 𝑑, where 𝑐 and 𝑑 are integers. By applying this property, we can take the cube root of each of these two factors separately to obtain the cube root of 𝑐 cubed times the cube root of 𝑑.

Since the cube root of 𝑐 cubed is 𝑐, this simplifies to give 𝑐 times the cube root of 𝑑. If we choose 𝑑 to be positive, then we can keep applying this process to take out all of the cube factors of 𝑑 until there are no more cube factors to take out.

At this point, we will have an expression in the form π‘Ž times the cube root of 𝑏, where π‘Ž and 𝑏 are integers and 𝑏 is the smallest possible positive value. To apply this process to the cube root of 32, we need to find the cube factors of 32. One way to do this is to factor 32 into primes. We see that 32 is equal to two to the fifth power. We see that the only nontrivial integer cube factor of this expression is two cubed. So we will rewrite 32 as two cubed times two squared.

We can then split the cube root over each of the factors separately to get the cube root of two cubed times the cube root of two squared. We can then calculate that the cube root of two cubed is two and that two squared is four. It is worth noting that four only has one as an integer cubic factor, so we cannot apply this process any further.

Hence, we were able to write the cube root of two times the cube root of 16 in the desired form with π‘Ž equal to two and 𝑏 equal to four, that is, two times the cube root of four.

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