Question

$\sqrt[3]{11025} \times \sqrt[3]{2835}$
​

Answer 1

Answer:

(11025×2835)^1/3

= (5×5×7×7×9 × 5×9×9×7) ^ 1/3

= (5^3 × 7^3 × 9^3) ^ 1/3

= [(315)^3]^1/3

= 315

Step-by-step explanation:

Answer 2

Answer:

$\sqrt[3]{11025}\sqrt[3]{2835}=315$

Step-by-step explanation:

$\sqrt[3]{11025}\sqrt[3]{2835}$

$\gray{\mathrm{Factor\:integer\:}11025=3^2\times \:5^2\times \:7^2}$

$=\sqrt[3]{3^2\times \:5^2\times \:7^2}\sqrt[3]{2835}$

$\gray{\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}}$

$\gray{\sqrt[3]{3^2\times \:5^2\times \:7^2}=\sqrt[3]{3^2}\sqrt[3]{5^2}\sqrt[3]{7^2}}$

$=\sqrt[3]{3^2}\sqrt[3]{5^2}\sqrt[3]{7^2}\sqrt[3]{2835}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}}$

$\gray{\sqrt[3]{3^2}=3^{2\times \dfrac{1}{3}},\:\sqrt[3]{5^2}=5^{2\times \dfrac{1}{3}},\:\sqrt[3]{7^2}=7^{2\times \dfrac{1}{3}}}$

$=3^{2\times \dfrac{1}{3}}\times \:5^{2\times \dfrac{1}{3}}\times \:7^{2\times \dfrac{1}{3}}\sqrt[3]{2835}$

$\gray{\mathrm{Refine}}$

$=3^{\dfrac{2}{3}}\times \:5^{\dfrac{2}{3}}\times \:7^{\dfrac{2}{3}}\sqrt[3]{2835}$

$\gray{\mathrm{Factor\:integer\:}2835=3^4\times \:5\times \:7}$

$=3^{\dfrac{2}{3}}\times \:5^{\dfrac{2}{3}}\times \:7^{\dfrac{2}{3}}\sqrt[3]{3^4\times \:5\times \:7}$

$\gray{\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}}$

$\gray{\sqrt[3]{3^4\times \:5\times \:7}=\sqrt[3]{3^4}\sqrt[3]{5}\sqrt[3]{7}}$

$=3^{\dfrac{2}{3}}\times \:5^{\dfrac{2}{3}}\times \:7^{\dfrac{2}{3}}\sqrt[3]{3^4}\sqrt[3]{5}\sqrt[3]{7}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}}$

$\gray{\sqrt[3]{3^4}=3^{4\times \dfrac{1}{3}}}$

$=3^{\dfrac{2}{3}}\times \:5^{\dfrac{2}{3}}\times \:7^{\dfrac{2}{3}}\times \:3^{4\times \dfrac{1}{3}}\sqrt[3]{5}\sqrt[3]{7}$

$\gray{\mathrm{Refine}}$

$=3^{\dfrac{2}{3}}\times \:5^{\dfrac{2}{3}}\times \:7^{\dfrac{2}{3}}\times \:3^{\dfrac{4}{3}}\sqrt[3]{5}\sqrt[3]{7}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \:a^b\times \:a^c=a^{b+c}}$

$\gray{3^{\dfrac{2}{3}}\times \:3^{\dfrac{4}{3}}=\:3^{\dfrac{2}{3}+\dfrac{4}{3}}}$

$=5^{\dfrac{2}{3}}\times \:7^{\dfrac{2}{3}}\times \:3^{\dfrac{2}{3}+\dfrac{4}{3}}\sqrt[3]{5}\sqrt[3]{7}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \:a^b\times \:a^c=a^{b+c}}$

$\gray{5^{\dfrac{2}{3}}\sqrt[3]{5}=\:5^{\dfrac{2}{3}}\times \:5^{\dfrac{1}{3}}=\:5^{\dfrac{2}{3}+\dfrac{1}{3}}}$

$=7^{\dfrac{2}{3}}\times \:3^{\dfrac{2}{3}+\dfrac{4}{3}}\times \:5^{\dfrac{2}{3}+\dfrac{1}{3}}\sqrt[3]{7}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \:a^b\times \:a^c=a^{b+c}}$

$\gray{7^{\dfrac{2}{3}}\sqrt[3]{7}=\:7^{\dfrac{2}{3}}\times \:7^{\dfrac{1}{3}}=\:7^{\dfrac{2}{3}+\dfrac{1}{3}}}$

$=3^{\dfrac{2}{3}+\dfrac{4}{3}}\times \:5^{\dfrac{2}{3}+\dfrac{1}{3}}\times \:7^{\dfrac{2}{3}+\dfrac{1}{3}}$

$\gray{\mathrm{Apply\:exponent\:rule}:\quad \:a^mb^m=\left(ab\right)^m}$

$\gray{5^{\dfrac{2}{3}+\dfrac{1}{3}}\times \:7^{\dfrac{2}{3}+\dfrac{1}{3}}=\left(5\times \:7\right)^{\dfrac{2}{3}+\dfrac{1}{3}}}$

$=3^{\dfrac{2}{3}+\dfrac{4}{3}}\left(5\times \:7\right)^{\dfrac{2}{3}+\dfrac{1}{3}}$

$\gray{\mathrm{Multiply\:the\:numbers:}\:5\times \:7=35}$

$=3^{\dfrac{2}{3}+\dfrac{4}{3}}\times \:35^{\dfrac{2}{3}+\dfrac{1}{3}}$

$\gray{3^{\dfrac{2}{3}+\dfrac{4}{3}}=3^2}$

$=3^2\times \:35^{\dfrac{2}{3}+\dfrac{1}{3}}$

$\gray{35^{\dfrac{2}{3}+\dfrac{1}{3}}=35}$

$=3^2\times \:35$

$\gray{3^2=9}$

$=9\times \:35$

$\gray{\mathrm{Multiply\:the\:numbers:}\:9\times \:35=315}$

$=315$