8th Class Mathematics Cubes and Cube Roots Cube Root of a Negative Number

         Cube Root of a Negative Number

 

The cube root of a negative number is always negative i.e. \[{{(-n)}^{\frac{1}{3}}}=-{{(n)}^{\frac{1}{3}}}\]

The cube root of -1000 is -10 because \[{{(-10)}^{3}}\]\[=-10\times -10\times -10=-1000\].

In symbolic form, the cube root of -1000 is written as \[\sqrt[3]{-1000}\]

So, \[\sqrt[3]{-1000}=-10\]

\[\because \]\[{{(-10)}^{3}}=-10\times -10\times -10=-1000\]  

 

From the above, we can infer that:

• The cube root of a positive number is a positive number.
• The cube root of a negative number is a negative number. In general:

• If \[\sqrt[3]{x}=a\] then, \[{{a}^{3}}=x\] where represents the cube root of \[x\].    

 

 

 

 

• If a number is divisible by 3, then its cube has digital root 9.
• If the remainder of the number is 1 when divide by 3, then its cube has digital root 1.
• If a number when divided by 3 leaves remainder 2, then its cube has digital root 8.
• Every positive rational number can be expressed as the sum of three positive rational cubes.    

 

 

 

 

• For any positive integer \[\sqrt[3]{-a}=-\sqrt[3]{a}\].
• The cube root of a number m is the number whose cube is m.
• The cube of a number is always raised to the power of three of that number.
• The cube of a even number is always even.
• The cube of odd number is always odd.
• The cube root of a number can be found by using the prime factorization methods.    

 

 

 

 

  Find the unit digit in the cube of the number 3331.

(a) 1                                                      

(b) 8

(c) 4                                                      

(d) 9

(e) None of these  

 

Answer: (a)  

Explanation:

We know that, \[{{\text{(3331)}}^{\text{3}}}=\text{3331}\times \text{3331}\times \text{3331}=\text{36959313691}\].    

 

 

  The smallest number by which 2560 must be multiplied so that the product will be a perfect cube.

(a) 35                                                    

(b) 25

(c) 8                                                      

(d) 5

(e) None of these

 

Answer: (b)

Explanation:

The factors of 2560 is given by

\[\text{256}0=\text{5}\times \text{8}\times \text{8}\times \text{8}\]

In this factors there are three 8 and one 5. So in order to make the number 5 perfect cube we have to multiply it by 25. Therefore, 25 is the least number by which it must be multiplied so that it becomes a perfect cube.  

 

 

  The smallest number by which we must divide 8788 so that it becomes a perfect cube.

(a) 2                                                      

(b) 169

(c) 4                                                      

(d) 13

(e) None of these  

 

Answer: (c)  

 

 

  Find the value of \[{{\left[ {{({{5}^{2}}+{{12}^{2}})}^{\frac{1}{2}}} \right]}^{3}}\] is given by:

(a) 2197                                               

(b) 169

(c) 1693                                               

(d) 289

(e) None of these  

 

Answer: (a)  

 

 

  Find the cube root of 42875.

(a) 35                                                    

(b) 25

(c) 15                                                    

(d) 20

(e) 32  

 

Answer: (a)

Explanation:

The factors of \[\text{42875}=\text{5}\times \text{5}\times \text{5}\times \text{7}\times \text{7}\times \text{7}\]

\[\sqrt[3]{42874}=\sqrt[3]{\underline{5\times 5\times 5}\times \underline{7\times 7\times 7}}=5\times 7=35\]

Make the factors of number by taking three identical numbers. Now multiply each number of the factors.

 

 

  Find the least number by which 3087 must be multiplied to make it a perfect cube.

(a) 3                                                      

(b) 4

(c) 9                                                      

(d) 7

(e) None of these  

 

Answer: (a)

Exploration:

The factors of \[\text{3}0\text{87}=\text{3}\times \text{3}\times \text{7}\times \text{7}\times \text{7}\]

We note that the factor 3 appears only 2 times, so if we multiply 3087 by 3 we get \[3084\times 3={{(3)}^{3}}\times {{(7)}^{3}}={{(3\times 7)}^{3}}\], so the smallest number is 3 which when multiplied to 3087, it gives a perfect cube.  

 

 

  Find the cube root of \[\frac{-2197}{1331}\].

(a) \[\frac{-13}{11}\]                                      

(b) \[\frac{13}{11}\]

(c) \[-\frac{13}{21}\]                                      

(d) \[-\frac{17}{21}\]

(e) None of these    

 

Answer: (a)  

 

 

  Find the value of \[\sqrt[3]{\frac{0.027}{0.008}}-\sqrt{\frac{0.09}{0.04}}-1\]

(a) 1                                                      

(b) -1

(c) 0                                                      

(d) \[\frac{3}{2}\]

(e) None of these  

 

Answer: (b)