Question Video: Simplifying the Addition of Cube Roots Mathematics

Express −5 ∛192 + 5 ∛−648 + ∛375 in its simplest form.

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Video Transcript

Express negative five multiplied by the cube root of 192 plus five multiplied by the cube root of negative 648 plus the cube root of 375 in its simplest form.

We will begin this question by simplifying each of the three terms separately. We will do this by using the fact that for any real numbers 𝑎 and 𝑏, the cube root of 𝑎 multiplied by the cube root of 𝑏 is equal to the cube root of 𝑎 multiplied by 𝑏. The largest perfect cube divisor of 192 is 64. Therefore, the cube root of 192 is equal to the cube root of 64 multiplied by three. This can be written as the cube root of 64 multiplied by the cube root of three, which in turn is equal to four multiplied by the cube root of three. The first term in our initial expression is therefore equal to negative five multiplied by four multiplied by the cube root of three.

We can use the same process to simplify the cube root of negative 648. This time, since negative 216 multiplied by three is equal to negative 648, we have the cube root of negative 216 multiplied by three. Using the product of cube roots property and our knowledge of perfect cubes, we have negative six multiplied by the cube root of three. The second term in our initial expression is therefore equal to five multiplied by this.

Next, we can rewrite the cube root of 375 as the cube root of 125 multiplied by the cube root of three. And this is equal to five multiplied by the cube root of three. We are now in a position where we can simplify each of the first two terms. The entire expression simplifies to negative 20 multiplied by the cube root of three minus 30 multiplied by the cube root of three plus five multiplied by the cube root of three. Finally, since negative 20 minus 30 plus five is equal to negative 45, we can conclude that the initial expression written in its simplest form is negative 45 multiplied by the cube root of three.

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