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What is the cube root of -216?

Answer
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Hint: Cube root of a number is a value which when multiplied by itself thrice or three times produces the original value. To find the cube root of -216, that is, 2163 , we have to split this as (1)×2163 . By using abn=an×bn , we can write (1)×2163=13×2163 . We have to find the cube root of -1 and 216 separately and substitute it in this form.

Complete step by step solution:
We have to find the cube root of -216. We know that the cube root of a number is a value which when multiplied by itself thrice or three times produces the original value. Cube root is denoted as 3 . Let us consider the cube root ofm3 , that is, m33 . We know that m×m×m=m3 . Hence, the cube root of m3 will be m.
Now, let us consider 2163 . We can write this as (1)×2163 .
We know that abn=an×bn . Hence, we can write (1)×2163 as
(1)×2163=13×2163...(i)
Let us first find the cube root of -1. We know that 1×1×1=1 . Therefore, we can write
13=1...(ii)
Now, let us find the cube root of 216. For this, let us find the prime factors of 216.
2|2162|1082|543|273|93|3 1
We can write 216 as 2×2×2×3×3×3 . We have to combine each triple.
216=2×2×2×3×3×3
Hence, we can write 2163=2×3=6...(iii) .
Let us substitute (ii) and (iii) in (i).
2163=1×6=6
Hence, the cube root of -216 is -6.

Note: Students must know the properties of roots and how to find the prime factors of 216. We can also denote cube root of a number, say m, as m to the power of 13 or m13 .
Therefore, we can write 2163=(216)13.

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