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Question

Find the cube root of each of the following rational numbers:
(i) -125729
(ii) 1064812167
(iii) -1968324389
(iv) 686-3456
(v) -39304-42875

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Solution

(i)
Let us consider the following rational number:

-125729

Now

-1257293
=-12537293 ( ab3=a3b3)
=-12537293 ( -a3=-a3 )
=-59 ( 729=9×9×9 and 125 = 5×5×5)

(ii)
Let us consider the following rational number:

1064812167

Now

10648121673
=106483121673 ( ab3=a3b3)

Cube root by factors:

On factorising 10648 into prime factors, we get:
10648=2×2×2×11×11×11

On grouping the factors in triples of equal factors, we get:
10648=2×2×2×11×11×11

Now, taking one factor from each triple, we get:
106483=2×11=22

Also

On factorising 12167 into prime factors, we get:
12167=23×23×23

On grouping the factors in triples of equal factors, we get:
12167=23×23×23

Now, taking one factor from the triple, we get:
121673=23

Now

10648121673
=106483121673
=2223


(iii)
Let us consider the following rational number:

-1968324389
Now,

-19683243893
=-196833243893 ( ab3=a3b3)
=-196833243893 ( -a3=-a3 )

Cube root by factors:

On factorising 19683 into prime factors, we get:
19683=3×3×3×3×3×3×3×3×3

On grouping the factors in triples of equal factors, we get:
19683=3×3×3×3×3×3×3×3×3

Now, taking one factor from each triple, we get:
196833=3×3×3=27

Also

On factorising 24389 into prime factors, we get:
24389=29×29×29

On grouping the factors in triples of equal factors, we get:
24389=29×29×29

Now, taking one factor from each triple, we get:
243893=29

Now

-19683243893
=-196833243893
=-196833243893
=-2729

(iv)
Let us consider the following rational number:

686-3456
Now

686-34563
=-2×7327×333 (686 and 3456 are not perfect cubes; therefore, we simplify it as 6863456 by prime factorisation.)
=-7326×333
=-73326×333=-723×23×333=-72×2×3=-712 ( ab3=a3b3)

(v)
Let us consider the following rational number:

-39304-42875
Now

-39304-428753
=-393043-428753 ( ab3=a3b3)
=-393043-428753 ( -a3=-a3 )

Cube root by factors:

On factorising 39304 into prime factors, we get:
39304=2×2×2×17×17×17

On grouping the factors in triples of equal factors, we get:
39304=2×2×2×17×17×17

Now, taking one factor from each triple, we get:
393043=2×17=34

Also

On factorising 42875 into prime factors, we get:
42875=5×5×5×7×7×7

On grouping the factors in triples of equal factors, we get:
42875=5×5×5×7×7×7

Now, taking one factor from each triple, we get:
428753=5×7=35

Now

-39304-428753
=-393043-428753
=-393043-428753
=-34-35=3435

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