Question

1. \\(sqrt{18x^2y^2}\\)\
2. \\((375x^{10}y^{13})^{\frac{1}{3}}\\)

Explanation:

Problem 11: Simplify $\boldsymbol{\sqrt{18x^2y^2}}$

Step1: Factor the radicand

Factor 18 into $9\times2$, and $x^2y^2$ is already a perfect square. So we have $\sqrt{9\times2\times x^2\times y^2}$.

Step2: Apply square - root properties

Using the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ ($a\geq0,b\geq0$) and $\sqrt{a^2}=|a|$ (for real numbers), we get $\sqrt{9}\times\sqrt{2}\times\sqrt{x^2}\times\sqrt{y^2}$. Since $x^2$ and $y^2$ are non - negative (as squares of real numbers), $\sqrt{x^2}=|x|$ and $\sqrt{y^2}=|y|$, and $\sqrt{9} = 3$. So the expression simplifies to $3|x||y|\sqrt{2}$. If we assume $x$ and $y$ are non - negative (a common assumption in many algebraic problems unless stated otherwise), then $|x| = x$ and $|y| = y$, and the expression is $3xy\sqrt{2}$.

Problem 13: Simplify $\boldsymbol{(375x^{10}y^{13})^{\frac{1}{3}}}$

Step1: Apply the power - of - a - product property

Using the property $(ab)^n=a^n\times b^n$, we can rewrite the expression as $375^{\frac{1}{3}}\times(x^{10})^{\frac{1}{3}}\times(y^{13})^{\frac{1}{3}}$.

Step2: Simplify each term

• For the coefficient: We know that $375 = 125\times3$, and $125 = 5^3$. So $375^{\frac{1}{3}}=(5^3\times3)^{\frac{1}{3}}$. Using the property $(ab)^n=a^n\times b^n$ again, we get $(5^3)^{\frac{1}{3}}\times3^{\frac{1}{3}}$. And by the property $(a^m)^n=a^{mn}$, $(5^3)^{\frac{1}{3}}=5^{3\times\frac{1}{3}} = 5$. So the coefficient simplifies to $5\times3^{\frac{1}{3}}=\sqrt[3]{125}\times\sqrt[3]{3}=\sqrt[3]{375}$ or $5\sqrt[3]{3}$.
• For the $x$ - term: Using the property $(a^m)^n=a^{mn}$, $(x^{10})^{\frac{1}{3}}=x^{10\times\frac{1}{3}}=x^{\frac{10}{3}}=x^{3+\frac{1}{3}}$. We can rewrite $x^{\frac{10}{3}}$ as $x^3\times x^{\frac{1}{3}}=\sqrt[3]{x}\times x^3$ (or just leave it as $x^{\frac{10}{3}}$).
• For the $y$ - term: Using the property $(a^m)^n=a^{mn}$, $(y^{13})^{\frac{1}{3}}=y^{13\times\frac{1}{3}}=y^{\frac{13}{3}}=y^{4+\frac{1}{3}}$. We can rewrite $y^{\frac{13}{3}}$ as $y^4\times y^{\frac{1}{3}}=\sqrt[3]{y}\times y^4$ (or just leave it as $y^{\frac{13}{3}}$).

Putting it all together, $(375x^{10}y^{13})^{\frac{1}{3}}=5\sqrt[3]{3}x^{\frac{10}{3}}y^{\frac{13}{3}}$ (or $5x^{3}\sqrt[3]{x}y^{4}\sqrt[3]{y}$ which can be further combined as $5x^{3}y^{4}\sqrt[3]{xy}$).

Answer:

s:

• For problem 11: If $x\geq0,y\geq0$, the answer is $\boldsymbol{3xy\sqrt{2}}$; in general form, it is $\boldsymbol{3|x||y|\sqrt{2}}$.
• For problem 13: The answer is $\boldsymbol{5\sqrt[3]{3}x^{\frac{10}{3}}y^{\frac{13}{3}}}$ (or $\boldsymbol{5x^{3}y^{4}\sqrt[3]{xy}}$).