Answer:
[tex]\sqrt[3]{2y^3} * 7\sqrt{18y} = 21(y^{\frac{3}{2}})(2^{\frac{5}{6}})[/tex]
Step-by-step explanation:
The question is poorly formatted.
Given
[tex]\sqrt[3]{2y^3} * 7\sqrt{18y}[/tex]
Required
Derive an equivalent expression
[tex]\sqrt[3]{2y^3} * 7\sqrt{18y}[/tex]
Express 18 as 9 * 2
[tex]\sqrt[3]{2y^3} * 7\sqrt{9 * 2y}[/tex]
Split the expression as follows:
[tex]\sqrt[3]{2y^3} * 7\sqrt{9} * \sqrt{2y}[/tex]
Take positive square root of 9
[tex]\sqrt[3]{2y^3} * 7*3 * \sqrt{2y}[/tex]
[tex]\sqrt[3]{2y^3} * 21 * \sqrt{2y}[/tex]
[tex]21*\sqrt[3]{2y^3} * \sqrt{2y}[/tex]
The cube root can be rewritten to give:
[tex]21*\sqrt[3]{2}*\sqrt[3]{y^3} * \sqrt{2y}[/tex]
[tex]\sqrt[3]{y^3} = y^{3*\frac{1}{3}} = y[/tex]
So, we have:
[tex]21*\sqrt[3]{2} * y * \sqrt{2y}[/tex]
Rewrite as:
[tex]21y *\sqrt[3]{2} * \sqrt{2y}[/tex]
Split [tex]\sqrt{2y[/tex]
[tex]21y *\sqrt[3]{2} * \sqrt{2} * \sqrt{y}[/tex]
Collect Like Terms
[tex]21y*\sqrt{y} *\sqrt[3]{2} * \sqrt{2}[/tex]
Represent in index form
[tex]21y*y^{\frac{1}{2}} *2^\frac{1}{3} *2^\frac{1}{2}[/tex]
Apply law of indices
[tex]21*y^{1+\frac{1}{2}} *2^{\frac{1}{3} +\frac{1}{2} }[/tex]
[tex]21*y^{\frac{2+1}{2}} *2^{\frac{2+3}{6}}[/tex]
[tex]21*y^{\frac{3}{2}} *2^{\frac{5}{6}}[/tex]
[tex]21(y^{\frac{3}{2}})(2^{\frac{5}{6}})[/tex]
Hence:
[tex]\sqrt[3]{2y^3} * 7\sqrt{18y} = 21(y^{\frac{3}{2}})(2^{\frac{5}{6}})[/tex]
