Question

please answer correctly​

Answer 1

Step-by-step explanation:

As per the provided information in the question, We have :

• $\sf \frac{1}{4} \sqrt[3]{128}$
• $\frac{2}{3} \sqrt{32}$
• $\frac{1}{7} \sqrt[3]{1029}$
• $\frac{2}{3} \sqrt{405}$

We are asked to convert these into pure surds.

In order to convert them into pure surds, We need to multiply the radicand with the number (Before the root), with the power of the index. If there is nothing in index, We can assume it as 2.

$\longmapsto \rm \frac{1}{4} \sqrt[3]{128}$

Doing the process with the given expressions,

$\longmapsto \rm \sqrt{128 \times {\bigg( \dfrac{1}{4} \bigg)}^{3} }$

$\longmapsto \rm \sqrt{128 \times \dfrac{1}{64} }$

$\longmapsto \rm \sqrt{128 \times \dfrac{1}{64} }$

$\longmapsto \rm8 \sqrt{2} \: \dfrac{1}{8}$

$\longmapsto \rm \sqrt{2}$

━━━━━━━━━━━━━━━━━

$\longmapsto \rm \dfrac{2}{3} \sqrt{32}$

$\longmapsto \rm \sqrt{32 \times {\bigg( \dfrac{2}{3} \bigg )}^{2} }$

$\longmapsto \rm \sqrt{32} \sqrt{ \dfrac{4}{9} }$

$\longmapsto \rm 4\sqrt{2}\times \dfrac{2}{3}$

$\longmapsto \rm \dfrac{8 \sqrt{2} }{3}$

━━━━━━━━━━━━━━━━━

$\longmapsto \rm \dfrac{1}{7} \sqrt[3]{1029}$

$\longmapsto \rm \sqrt{1029 \: \times \bigg( {\frac{1}{7} \bigg)}^{3} }$

$\longmapsto \rm \sqrt{1029 \: \times \bigg( \dfrac{1}{343} \bigg )}$

$\longmapsto \rm \dfrac{1 \times 7 \sqrt{21}}{7 \sqrt{7} }$

$\longmapsto \rm \dfrac{1 \times \sqrt{21}}{\sqrt{7} }$

$\longmapsto \rm \dfrac{ \sqrt{21}}{\sqrt{7} }$

$\longmapsto \rm \sqrt{3}$

━━━━━━━━━━━━━━━━━

$\longmapsto \rm \dfrac{2}{3} \sqrt{405}$

$\longmapsto \rm \sqrt{405 \times \bigg( {\dfrac{2}{3} \bigg)}^{2}}$

$\longmapsto \rm \sqrt{405 \times \dfrac{4}{9} }$

$\longmapsto \rm \sqrt{405} \sqrt{\dfrac{4}{9} }$

$\longmapsto \rm \sqrt{405} \dfrac{2}{3}$

$\longmapsto \rm \dfrac{2\sqrt{405}}{3}$

$\longmapsto \rm \dfrac{2 \sqrt{5 \times {9}^{2} }}{3}$

$\longmapsto \rm \dfrac{2 \sqrt{5}\times \sqrt{{9}^{2} }}{3}$

$\longmapsto \rm \dfrac{2 \sqrt{5}\times {9}}{3}$

$\longmapsto \rm \dfrac{2 \sqrt{5}\times {3}}{1}$

$\longmapsto \rm 6 \sqrt{5}$

Answer 2

Step-by-step explanation:

As per the provided information in the question, We have :

\sf \frac{1}{4} \sqrt[3]{128}

4

1

3

128

\frac{2}{3} \sqrt{32}

3

2

32

\frac{1}{7} \sqrt[3]{1029}

7

1

3

1029

\frac{2}{3} \sqrt{405}

3

2

405

We are asked to convert these into pure surds.

In order to convert them into pure surds, We need to multiply the radicand with the number (Before the root), with the power of the index. If there is nothing in index, We can assume it as 2.

\longmapsto \rm \frac{1}{4} \sqrt[3]{128}⟼

4

1

3

128

Doing the process with the given expressions,

\longmapsto \rm \sqrt{128 \times {\bigg( \dfrac{1}{4} \bigg)}^{3} }⟼

128×(

4

1

)

3

\longmapsto \rm \sqrt{128 \times \dfrac{1}{64} }⟼

128×

64

1

\longmapsto \rm \sqrt{128 \times \dfrac{1}{64} }⟼

128×

64

1

\longmapsto \rm8 \sqrt{2} \: \dfrac{1}{8}⟼8

2

8

1

\longmapsto \rm \sqrt{2}⟼

2

━━━━━━━━━━━━━━━━━

\longmapsto \rm \dfrac{2}{3} \sqrt{32}⟼

3

2

32

\longmapsto \rm \sqrt{32 \times {\bigg( \dfrac{2}{3} \bigg )}^{2} }⟼

32×(

3

2

)

2

\longmapsto \rm \sqrt{32} \sqrt{ \dfrac{4}{9} }⟼

32

9

4

\longmapsto \rm 4\sqrt{2}\times \dfrac{2}{3}⟼4

2

×

3

2

\longmapsto \rm \dfrac{8 \sqrt{2} }{3}⟼

3

8

2

━━━━━━━━━━━━━━━━━

\longmapsto \rm \dfrac{1}{7} \sqrt[3]{1029}⟼

7

1

3

1029

\longmapsto \rm \sqrt{1029 \: \times \bigg( {\frac{1}{7} \bigg)}^{3} }⟼

1029×(

7

1

)

3

\longmapsto \rm \sqrt{1029 \: \times \bigg( \dfrac{1}{343} \bigg )}⟼

1029×(

343

1

)

\longmapsto \rm \dfrac{1 \times 7 \sqrt{21}}{7 \sqrt{7} }⟼

7

7

1×7

21

\longmapsto \rm \dfrac{1 \times \sqrt{21}}{\sqrt{7} }⟼

7

1×

21

\longmapsto \rm \dfrac{ \sqrt{21}}{\sqrt{7} }⟼

7

21

\longmapsto \rm \sqrt{3}⟼

3

━━━━━━━━━━━━━━━━━

\longmapsto \rm \dfrac{2}{3} \sqrt{405}⟼

3

2

405

\longmapsto \rm \sqrt{405 \times \bigg( {\dfrac{2}{3} \bigg)}^{2}}⟼

405×(

3

2

)

2

\longmapsto \rm \sqrt{405 \times \dfrac{4}{9} }⟼

405×

9

4

\longmapsto \rm \sqrt{405} \sqrt{\dfrac{4}{9} }⟼

405

9

4

\longmapsto \rm \sqrt{405} \dfrac{2}{3}⟼

405

3

2

\longmapsto \rm \dfrac{2\sqrt{405}}{3}⟼

3

2

405

\longmapsto \rm \dfrac{2 \sqrt{5 \times {9}^{2} }}{3}⟼

3

2

5×9

2

\longmapsto \rm \dfrac{2 \sqrt{5}\times \sqrt{{9}^{2} }}{3}⟼

3

2

5

×

9

2

\longmapsto \rm \dfrac{2 \sqrt{5}\times {9}}{3}⟼

3

2

5

×9

\longmapsto \rm \dfrac{2 \sqrt{5}\times {3}}{1}⟼

1

2

5

×3

\longmapsto \rm 6 \sqrt{5}⟼6

5