RS Aggarwal solution class 8 chapter 4 Cubes and Cube roots Exercise 4C

Exercise 4C

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Q1 | Ex-4C | Class 8 | RS AGGARWAL | Cubes and Cube roots | Chapter 4 | myhelper

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Question 1:

Evaluate:
$\sqrt[3]{64}$

Answer 1:

$\sqrt[3]{64}$
By prime factorisation:
64 = $2\times 2\times 2\times 2\times 2\times 2$
     = $\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)$

∴ $\sqrt[3]{64}=\sqrt[3]{(2{)}^3\times (2{)}^3}=\left(2\times 2\right)=4$

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Question 2:

Evaluate:
$\sqrt[3]{343}$

Answer 2:

$\sqrt[3]{343}$
By prime factorisation:
343 = $7\times 7\times 7$
      = ( $7\times 7\times 7$ )

∴ $\sqrt[3]{343}=\sqrt[3]{7^3}=7$

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Question 3:

Evaluate:
$\sqrt[3]{729}$

Answer 3:

$\sqrt[3]{729}$
By prime factorisation:

729 = $3\times 3\times 3\times 3\times 3\times 3$
       = $\left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)$

∴ $\sqrt[3]{729}$ = $\left(3\times 3\right)= 9$

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Question 4:

Evaluate:
$\sqrt[3]{1728}$

Answer 4:

$\sqrt[3]{1728}$
By prime factorisation:

1728 = $2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3$
         = $\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times \left(3\times 3\times 3\right)=2^3\times 2^3\times 3^3$

∴ $\sqrt[3]{1728}$ = $\left(2\times 2\times 3\right)= 12$

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Question 5:

Evaluate:
$\sqrt[3]{9261}$

Answer 5:

$\sqrt[3]{9261}$
By prime factorisation:

9261 = $3\times 3\times 3\times 7\times 7\times 7$
         = $\left(3\times 3\times 3\right)\times \left(7\times 7\times 7\right)=3^3\times 7^3$

∴ $\sqrt[3]{9261}$ = $\left(3\times 7\right) = 21$

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Question 6:

Evaluate:
$\sqrt[3]{4096}$

Answer 6:

$\sqrt[3]{4096}$
By prime factorisation:

4096 = $2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$
         = $$\begin{gathered} \left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right) \\ =2^3\times 2^3\times 2^3\times 2^3 \end{gathered}$$

∴ $\sqrt[3]{4096}$ = $\left(2\times 2\times 2\times 2\right) = 16$

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Question 7:

Evaluate:
$\sqrt[3]{8000}$

Answer 7:

$\sqrt[3]{8000}$
By prime factorisation:

8000 = $2\times 2\times 2\times 2\times 2\times 2\times 5\times 5\times 5$
         = $\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times \left(5\times 5\times 5\right)$

∴ $\sqrt[3]{8000}$ = $\left(2\times 2\times 5\right) = 20$

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Question 8:

Evaluate:
$\sqrt[3]{3375}$

Answer 8:

$\sqrt[3]{3375}$
By prime factorisation:

3375 = $3\times 3\times 3\times 5\times 5\times 5$
         = $\left(3\times 3\times 3\right)\times \left(5\times 5\times 5\right)$

∴ $\sqrt[3]{3375}$ = $\left(3\times 5\right) = 15$

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Question 9:

Evaluate:
$\sqrt[3]{-216}$

Answer 9:

$\sqrt[3]{-216}$
By prime factorisation:

216 = $2\times 2\times 2\times 3\times 3\times 3$
        = $\left(2\times 2\times 2\right)\times \left(3\times 3\times 3\right)$
$\sqrt[3]{-216}$ = $-\left(2\times 3\right) = -6$

∴ $\sqrt[3]{-216}$ =  $-\left(\sqrt[3]{216}\right)=-6$

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Question 10:

Evaluate:
$\sqrt[3]{-512}$

Answer 10:

$\sqrt[3]{-512}$
By prime factorisation:

= $2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$
             = $\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)$
$\sqrt[3]{-512}$ =

∴$\sqrt[3]{-512}$ = $-\left(\sqrt[3]{512}\right)=-8$

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Question 11:

Evaluate:
$\sqrt[3]{-1331}$

Answer 11:

$\sqrt[3]{-1331}$
By prime factorisation:
$\sqrt[3]{1331}$ = $\sqrt[3]{11\times 11\times 11}$

             
$\sqrt[3]{-1331}$= $-{\left(11\times 11\times 11\right)}^{\frac{1}{3}} = -11$
∴ $\sqrt[3]{-1331}=-\left(\sqrt[3]{1331}\right)=-11$ 

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Question 12:

Evaluate:
$\sqrt[3]{\frac{27}{64}}$

Answer 12:

$\sqrt[3]{\frac{27}{64}}$
By prime factorisation:
                 

$\sqrt[3]{\frac{27}{64}}$ = $\frac{\sqrt[3]{27}}{\sqrt[3]{64}}$= $\frac{\sqrt[3]{\left(3\times 3\times 3\right)}}{\sqrt[3]{\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)}}$= $\frac{\sqrt[3]{\left(3\times 3\times 3\right)}}{\sqrt[3]{\left(4\times 4\times 4\right)}}=\frac{3}{4}$
∴ $\sqrt[3]{\frac{27}{64}}$= $\frac{3}{4}$

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Question 13:

Evaluate:
$\sqrt[3]{\frac{125}{216}}$

Answer 13:

$\sqrt[3]{\frac{125}{216}}$
By prime factorisation:

                

$\sqrt[3]{\frac{125}{216}}$  = $\frac{\sqrt[3]{5\times 5\times 5}}{\sqrt[3]{\left(2\times 2\times 2\right)\times \left(3\times 3\times 3\right)}}$ = $\frac{\sqrt[3]{5\times 5\times 5}}{\sqrt[3]{\left(6\times 6\times 6\right)}}=\frac{5}{6}$

∴ $\sqrt[3]{\frac{125}{216}}$ = $\frac{5}{6}$

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Question 14:

Evaluate:
$\sqrt[3]{\frac{-27}{125}}$

Answer 14:

$\sqrt[3]{\frac{-27}{125}}$
            
By factorisation:
=

∴ $\sqrt[3]{\frac{-27}{125}}$= $\frac{-3}{5}$

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Question 15:

Evaluate:
$\sqrt[3]{\frac{-64}{343}}$

Answer 15:

$\sqrt[3]{\frac{-64}{343}}$
On factorisation:
                

=
∴ $\sqrt[3]{\frac{-64}{343}}$= $\frac{-4}{7}$

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Question 16:

Evaluate:
$\sqrt[3]{64\times 729}$

Answer 16:

$\sqrt[3]{64\times 729}$
$\sqrt[3]{64\times 729}$ = $\sqrt[3]{64}\times \sqrt[3]{729}$
                 = $\sqrt[3]{4\times 4\times 4}$ $\times \sqrt[3]{\left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)}$
                 = $\sqrt[3]{4\times 4\times 4}$ $\times \sqrt[3]{\left(9\times 9\times 9\right)}$
$\sqrt[3]{64\times 729}$ = $\left(4\right)\times \left(9\right)$= 36

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Question 17:

Evaluate:
$\sqrt[3]{\frac{729}{1000}}$

Answer 17:

$\sqrt[3]{\frac{729}{1000}}$
             
On factorisation:
$\sqrt[3]{\frac{729}{1000}}$= $\frac{\sqrt[3]{\left(3\times 3\times 3\right)\times \left(3\times 3\times 3\right)}}{\sqrt[3]{\left(2\times 2\times 2\right)\times \left(5\times 5\times 5\right)}}$= $\frac{\sqrt[3]{9\times 9\times 9}}{\sqrt[3]{10\times 10\times 10}}$
$\sqrt[3]{\frac{729}{1000}}$= $\frac{9}{10}$

Q18 | Ex-4C | Class 8 | RS AGGARWAL | Cubes and Cube roots | Chapter 4 | myhelper

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Question 18:

Evaluate:
$\sqrt[3]{\frac{-512}{343}}$

Answer 18:

$\sqrt[3]{\frac{-512}{343}}$
By factorisation:

                        

=
$\sqrt[3]{\frac{-512}{343}}$= $\frac{-8}{7}$