Question

Solve this questions without any spamming please..... If possible then please show it with explanation , don't write simply answers

Answer 1

Questions :

• Find out the cube roots of

1. (-1.2)
2. 5
3. -5/3
4. 0.03

• Write the numbers in their cubic forms

1. -1331
2. -0.000008
3. 729/-216
4. 15.625

• Find the smallest number to be multiplied to the number 2560 to get a perfect cube . Also find out the cube root of the number obtained.

• By which number should 8788 be divided to get a perfect cube ? Also find out the cube root of the number so obtained.

• Find cube roots of the following :

1. 5832
2. -474552

• Find the cube root through estimation method :

1. 10648
2. 91125

Answers :

• Cube roots of :

1. (-1.2)
2. 5
3. -5/3
4. 0.03

=> Cube root : The number when reduced 3 products by itself gives us a cube root.

1. $\sf{(-1.2)}$

$\longrightarrow{\sf{(-1.2)}}$

$\longrightarrow{\sf{\sqrt[3]{x} = {x}^{\dfrac{1}{3}}}}$

$\longrightarrow{\sf{{(-1.2)}^{\dfrac{1}{3}}}}$

$\implies{\sf{-1.062658...}}$

_________________

2. $\sf{5}$

$\longrightarrow{\sf{5}}$

$\longrightarrow{\sf{\sqrt[3]{x} = {x}^{\dfrac{1}{3}}}}$

$\longrightarrow{\sf{{(5)}^{\dfrac{1}{3}}}}$

$\implies{\sf{1.70997...}}$

__________________

3. $\sf{\dfrac{-5}{3}}$

$\longrightarrow{\sf{(\dfrac{-5}{3})}}$

$\longrightarrow{\sf{\sqrt[3]{x} = {x}^{\dfrac{1}{3}}}}$

$\longrightarrow{\sf{{(\dfrac{-5}{3})}^{\dfrac{1}{3}}}}$

$\implies{\sf{1.185631...}}$

___________________

4. $\sf{0.03}$

$\longrightarrow{\sf{(0.03)}}$

$\longrightarrow{\sf{\sqrt[3]{x} = {x}^{\dfrac{1}{3}}}}$

$\longrightarrow{\sf{{(0.03)}^{\dfrac{1}{3}}}}$

$\implies{\sf{0.31072325....}}$

_________________

• Write in cube form

1. -1331
2. -0.000008
3. 729/-216
4. 15.625

=> Cubic form: The number when written in the power of 3 or with a cubic power. The following answers are on the basis of prime factorization.

1. $\sf{(-1331)}$

$\longrightarrow{\sf{(-11) \times (-11) \times (-11) \implies (-1331)}}$

$\implies{\sf{\sqrt[3]{11} = (-1331)}}$

_________________

2. $\sf{(-0.000008)}$

$\longrightarrow{\sf{(-0.02) \times (-0.02) \times (-0.02) \implies (-0.000008)}}$

$\implies{\sf{\sqrt[3]{-0.02} = (-0.000008)}}$

_________________

3. $\sf{(\dfrac{729}{-216})}$

$\longrightarrow{\sf{(\dfrac{9}{-6}) \times (\dfrac{9}{-6}) \times (\dfrac{9}{-6}) \implies (\dfrac{729}{-216})}}$

$\implies{\sf{\sqrt[3]{\dfrac{9}{-6}} = (\dfrac{729}{-216})}}$

_________________

4. $\sf{(15.625)}$

$\longrightarrow{\sf{(2.5) \times (2.5) \times (2.5) \implies (15.625)}}$

$\implies{\sf{\sqrt[3]{2.5} = (15.625)}}$

• Find the smallest number to be multiplied to 2560 to get a perfect cube. Also write the cube root of the number so obtained.

=> Following the prime factorization method we find that 5 is left unpaired and we must multiply 5 * 5 with 2560 in order to get a perfect cube .

5* 5 = 25

25 * 2560 = 64000

$\longrightarrow{\sf{64000 = \sqrt[3]{40}}}$

_________________

• By which number should 8788 be divided to get a perfect cube ? Also find out the cube root of the number so obtained.

=> Following the prime factorization method we find out that 2 is left unpaired and hence we will divide 2*2 with 8788 to make it a perfect cube.

2*2 = 4

8788 ÷ 4 = 2197

$\longrightarrow{\sf{2197 = \sqrt[3]{13}}}$

__________________

• Find cube roots :

1. 5832
2. -474552

=>

1) By prime factorization we find :

2*2*2*3*3*3*3*3*3

So taking pairs = 2*3*3 => 18

$\longrightarrow{\sf{ 5832 = \sqrt[3]{18}}}$

_________________

2) By prime factorization we find :

-2*-2*-2*-3*-3*-3*-13*-13*-13

Taking as pairs = -2*-3*-13 =>-78

$\longrightarrow{\sf{-474552 = \sqrt[3]{-78}}}$

_________________

• Find cube roots through estimation method :

1. 10648
2. 91125

=>

1) Divide the number into two parts :

10 and 648

648 = 8 = $\sf{{2}^{3}}$

So the first number is 2.

Second part : 8 > 10 > 27

$\longrightarrow{\sf{{2}^{3} > {10}^{} > {3}^{3}}}$

So it's 22 by taking 2 twice.

_________________

2) Divide into two parts :

91 and 125

125 = $\sf{{5}^{3}}$

91 = 64 is nearest => $\sf{{4}^{3}}$

Highest cube in both = 91

So , the answer is 45 by taking 4 and 5 as the digits.

Answer 2

Find out the cubes of -

$\huge\rm\red { Question }$

• $\sf\red { - 1.2 }$

$\huge\rm\green { Solution }$

$\sf { ☞ ( - {1.2})^{3} = ( \frac{ \cancel { { - 12}} \: {}^{ - 6} }{ \cancel{10} {}^{ \: 5} } ) {}^{3} = ( \frac{ - 6}{5} ) {}^{ 3} }$

$\sf { ☞ \frac{ (- 6) \times ( - 6) \times ( - 6)}{ 5 \times5 \times 5} = \frac{ - 216}{125} = 1.728 }$

${\boxed {\bf {\red { ( - 1.2) {}^{3} = 1.728 \: }}}}$

_____________________

$\huge\rm\red { Question }$

• $\sf\red { 5}$

$\huge\rm\green { Solution }$

$\sf { ☞{(5)}^{3} = 5 \times 5 \times 5 = 125 }$

${\boxed {\bf {\red { ( 5) {}^{3} = 125}}}}$

______________________

$\huge\rm\red { Question }$

• $\sf\red { \frac{-5}{3} }$

$\huge\rm\green { Solution }$

$\sf {☞ ( { \frac{ - 5}{3} )}^{3} = \frac{( - 5) {}^{3} }{ { (3) }^{3} } }$

$\sf { ☞ \frac{( - 5) \times ( - 5) \times ( - 5)}{ 3 \times 3 \times 3} = \frac{ - 127}{27} = - 4.629...}$

${\boxed {\red {\bf { \frac{-5}{3}^{3} = \frac{ - 127}{27} }}}}$

____________________

$\huge\rm\red { Question }$

• $\sf\red { 0.03 }$

$\huge\rm\green { Solution }$

$\sf {☞ \: (0.03) {}^{3} = ( { \frac{3}{100} )}^{3} = \frac{ {(3)}^{3} }{(100) {}^{3} }}$

$\sf { ☞ \: \frac{3 \times3 \times 3}{100 \times 100 \times 100} = \frac{27}{1000000} = 0.000027}$

${\boxed {\bf {\red { {(0.03)}^{3} = 0.000027 }}}}$

_________....

Write down the following numbers in their cubic form -

$\huge\rm\red { Question }$

• $\sf\red { - 1331 }$

$\huge\rm\green { Solution }$

Resolving -1331 into prime factors,

⠀⠀⠀⠀⠀⠀⠀⠀⠀11 | 1331

⠀⠀⠀⠀⠀⠀⠀⠀⠀11 | 121

⠀⠀⠀⠀⠀⠀⠀⠀⠀11 | 11

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ | 1

By prime factorization we get -

$☞ \sf { - 1331 = \underline{ ( - 11) \times ( - 11) \times ( - 11)} }$

$☞ \sf { - 1331 = {( - 11)}^{3} }$

${\boxed {\red {\bf { -1331 = {( - 11)}^{3} }}}}$

__________________

$\huge\rm\red { Question }$

• $\sf\red { - 0.000008 }$

$\huge\rm\green { Solution }$

We know,

$\sf { -0.000008 = \frac{-8}{1000000} }$

So, we'll resolve both denominator and numerator into prime factors.

Resolving -8 and 1000000 into prime factors,

⠀⠀⠀⠀⠀⠀2 | 8⠀⠀⠀⠀⠀⠀⠀⠀⠀10 | 1000000

⠀⠀⠀⠀⠀⠀2 | 4⠀⠀⠀⠀⠀⠀⠀⠀⠀10 | 100000

⠀⠀⠀⠀⠀⠀2 | 2⠀⠀⠀⠀⠀⠀⠀⠀⠀10 | 10000

⠀⠀⠀⠀⠀⠀⠀ | 1⠀⠀⠀⠀⠀⠀⠀⠀⠀10 | 1000

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀10 | 100

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀10 | 10

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀1

By prime factorization we get-

$☞ \sf { - 8 = \underline { ( - 2) \times ( - 2) \times ( - 2)} = ( - 2) {}^{3} }$

and,

$☞ \sf { 1000000 = \underline{10 \times 10 \times 10} \times \underline {10 \times 10 \times 10} }$

$\sf {= {10 }^{3} × {10}^{3} = ( {100)}^{3} }$

So we can say that,

$\sf { - 0.000008 = \frac{ - 8}{1000000} = \frac{ {( - 2)}^{3} }{ {(100)}^{3}} ={( \frac{ - 2}{100} )}^{3} = {( - 0.02)}^{3} }$

${\boxed {\red {\bf { - 0.000008 = {( - 0.02)}^{3} }}}}$

_____________________

$\huge\rm\red { Question }$

• $\sf\red { \frac{729}{ - 216} }$

$\huge\rm\green { Solution }$

Again as the previous question we'll resolve both denominator and numerator in prime factors-

Resolving 729 and -216 in prime factors,

⠀⠀⠀⠀⠀3 | 729⠀⠀⠀⠀⠀⠀2 | 216

⠀⠀⠀⠀⠀3 | 243⠀⠀⠀⠀⠀⠀ 2 | 108

⠀⠀⠀⠀⠀3 | 81⠀⠀⠀⠀⠀⠀⠀ 2 | 54

⠀⠀⠀⠀⠀3 | 27⠀⠀⠀⠀⠀⠀⠀3 | 27

⠀⠀⠀⠀⠀3 | 9⠀⠀⠀⠀⠀⠀⠀ 3 | 9

⠀⠀⠀⠀⠀3 | 3⠀⠀⠀⠀⠀⠀⠀⠀3 | 3

⠀⠀⠀⠀⠀⠀| 1⠀⠀⠀⠀⠀⠀⠀⠀⠀ | 1

By prime factorization we get-

$☞ \sf { 729 = \underline{3 \times 3 \times 3} \times \underline {3 \times 3 \times 3} }$

$\sf { = {3}^{3}×{3}^{3}= {(9)}^{3} }$

and,

$☞ \sf { -216 = \underline - { 2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} }$

$\sf { = -{2}^{3}×{3}^{3}= {(-6)}^{3} }$

So we can say that,

$\sf \red { = \frac{729}{ - 216} = ( \frac{ 9}{ - 6} ) {}^{3} }$

${\boxed {\red {\bf { \frac{729}{ - 216} = ( \frac{ 9}{ - 6} ) {}^{3} }}}}$

_______________________

$\huge\rm\red { Question }$

• $\sf\red { 15.625 }$

$\huge\rm\green { Solution }$

We know,

$\sf { 15.625 = \frac{15625}{1000} }$

So, we'll resolve both denominator and numerator into prime factors.

Resolving 15625 and 1000 into prime factors,

⠀⠀⠀⠀⠀5 | 15625⠀⠀⠀⠀⠀10 | 1000

⠀⠀⠀⠀⠀5 | 3125⠀⠀⠀⠀⠀ 10 | 100

⠀⠀⠀⠀⠀5 | 625⠀⠀⠀⠀⠀⠀10 | 10

⠀⠀⠀⠀⠀5 | 125⠀⠀⠀⠀⠀⠀⠀⠀| 1

⠀⠀⠀⠀⠀5 | 25

⠀⠀⠀⠀⠀5 | 5

⠀⠀⠀⠀⠀ | 1

By prime factorization we get-

$☞ \sf { 15625= \underline{5 \times 5 \times 5} \times \underline {5 \times 5 \times 5} }$

$\sf { = {5}^{3}×{5}^{3}= {(25)}^{3} }$

and,

$☞ \sf { 1000= \underline{10 \times 10 \times 10} = {10}^{3}}$

So, we can say that

$\sf { 15.625 = ( \frac{25}{10} ) ^{3} = ( 2.5) {}^{3} }$

${\boxed {\red {\bf { 15.625 =(2.5) {}^{3} }}}}$