Question

State with reasons which of the following are surds and which are not:
$1. \sqrt[3]{4} \times 5 \sqrt[3]{16}$
$2. \sqrt[4]{3} \times \sqrt[3]{9}$
$3. \sqrt[5]{16} \div \sqrt[3]{32}$
$4.( \sqrt{2} + \sqrt{3} ) \times ( \sqrt{2} - \sqrt{3} )$
​

Answer 1

Answer:

(1)

$\sqrt[3]{4} \times 5 \sqrt[3]{16}$

$\sqrt[3]{2 \times 2} \times 5 \sqrt[3]{2 \times 2 \times 2 \times 2}$

$5 \times \sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2}$

$5 \times 2 \times 2 \: \: \: \:$

$20$

(2)

$\sqrt[4]{3} \times \sqrt[3]{9}$

${3}^{ \frac{1}{4} } \times {9}^{ \frac{1}{3} }$

${3}^{ \frac{1}{4} } \times {3}^{ \frac{1}{3} } \times {3}^{ \frac{1}{3} }$

${(3)}^{ \frac{1}{4} + \frac{1}{3} + \frac{1}{3} }$

$\: {3}^{ \frac{11}{12} }$

(3)

$\sqrt[5]{16} \div \sqrt[3]{32}$

${16}^{ \frac{1}{5} } \div {32}^{ \frac{1}{3} }$

${ ({(2)}^{4} )}^{ \frac{1}{5} } \div { ({(2)}^{5} )}^{ \frac{1}{3} }$

${2}^{ \frac{4}{5} } \div {2}^{ \frac{5}{3} }$

${(2)}^{ \frac{4}{5} - \frac{5}{3} }$

${(2)}^{ \frac{12 - 25}{15} }$

${(2)}^{ - \frac{13}{15} }$

$( { \frac{1}{2}) }^{ \frac{13}{15} }$

(4)

$( \sqrt{2} + \sqrt{3} ) \times ( \sqrt{2} - \sqrt{3} )$

${( \sqrt{2} \: ) }^{2} - {( \sqrt{3} \: )}^{2}$

$\: 4 - 9$

$\: \: - 5$