which expression is equivalent to $sqrt{8x^{7}y^{8}}$? assume $xgeq0$.\n$xy^{2}sqrt{8x^{3}}$\n$2x^{2}y^{2}sqr…

which expression is equivalent to $sqrt{8x^{7}y^{8}}$? assume $xgeq0$.\n$xy^{2}sqrt{8x^{3}}$\n$2x^{2}y^{2}sqrt{xy^{2}}$\n$2x^{3}y^{4}sqrt{2x}$\n$4x^{3}y^{4}sqrt{x}$

which expression is equivalent to $sqrt{8x^{7}y^{8}}$? assume $xgeq0$.\n$xy^{2}sqrt{8x^{3}}$\n$2x^{2}y^{2}sqrt{xy^{2}}$\n$2x^{3}y^{4}sqrt{2x}$\n$4x^{3}y^{4}sqrt{x}$

Answer

Explanation:

Step1: Rewrite the radicand

Rewrite $\sqrt{8x^{7}y^{8}}$ as $\sqrt{8}\times\sqrt{x^{7}}\times\sqrt{y^{8}}$.

Step2: Simplify each square - root

  • For $\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}$.
  • For $\sqrt{x^{7}}=\sqrt{x^{6}\times x}=x^{3}\sqrt{x}$ (since $\sqrt{a^{m}\times b}=\sqrt{a^{m}}\times\sqrt{b}$ and $\sqrt{a^{m}} = a^{\frac{m}{2}}$, here $m = 6$ for $x^{6}$).
  • For $\sqrt{y^{8}}=y^{4}$ (because $\sqrt{y^{8}}=y^{\frac{8}{2}} = y^{4}$).

Step3: Combine the simplified terms

$2\sqrt{2}\times x^{3}\sqrt{x}\times y^{4}=2x^{3}y^{4}\sqrt{2x}$.

Answer:

$2x^{3}y^{4}\sqrt{2x}$