Question

${625}^{\frac{1}{3}}\times {1600}^{\frac{1}{3}}$ = ___

• A. 50
• B. 75
• C. 100
• D. 125

Answer 1

The correct option is C 100

${625}^{\frac{1}{3}}\times {1600}^{\frac{1}{3}}=(625\times 1600{)}^{\frac{1}{3}}(\because a^m\times b^m=\left(ab{)}^m\right)$
$={\{{\left(25\right)}^2\times (4^3\times 5^2)\}}^{\frac{1}{3}}$
$={\{{\left(5^2\right)}^2\times 4^3\times 5^2\}}^{\frac{1}{3}}$
$={\{5^4\times 4^3\times 5^2\}}^{\frac{1}{3}}(\because (a^m{)}^n=a^{mn})$
$={\{5^{4+2}\times 4^3\}}^{\frac{1}{3}}(\because a^m\times a^n=a^{m+n})$
$=(5^6\times 4^3{)}^{\frac{1}{3}}$
$=(5^6{)}^{\frac{1}{3}}\times (4^3{)}^{\frac{1}{3}}(\because (a\times b{)}^m=a^m\times b^m)$
$=5^{6\times \frac{1}{3}}\times 4^{3\times \frac{1}{3}}(\because (a^m{)}^n=a^{mn})$
$=5^2\times 4$
$=25\times 4$
$=100$