Question

1. $\sqrt{18x^{4}y}$
2. $\sqrt{50x^{3}y^{4}}$
3. $\sqrt{192a^{3}b}$
4. $\sqrt{112x^{3}y^{4}}$
5. $\sqrt{216x^{3}y^{3}}$

Explanation:

Let's solve each problem one by one. We'll use the property of square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ (where $a \geq 0$ and $b \geq 0$) and simplify the radicand by factoring out perfect squares.

Problem 21: $\boldsymbol{\sqrt{18x^4y}}$

Step 1: Factor the radicand into perfect squares and other factors

Factor $18$ as $9 \times 2$, and $x^4$ is a perfect square ($(x^2)^2$). So we have:
$$\sqrt{18x^4y} = \sqrt{9 \times 2 \times x^4 \times y}$$

Step 2: Apply the square root property

Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get:
$$\sqrt{9} \times \sqrt{x^4} \times \sqrt{2y}$$
Since $\sqrt{9} = 3$ and $\sqrt{x^4} = x^2$, this simplifies to:
$$3x^2\sqrt{2y}$$

Problem 23: $\boldsymbol{\sqrt{50x^3y^4}}$

Step 1: Factor the radicand

Factor $50$ as $25 \times 2$, $x^3$ as $x^2 \times x$, and $y^4$ is a perfect square ($(y^2)^2$). So:
$$\sqrt{50x^3y^4} = \sqrt{25 \times 2 \times x^2 \times x \times y^4}$$

Step 2: Apply the square root property

$$\sqrt{25} \times \sqrt{x^2} \times \sqrt{y^4} \times \sqrt{2x}$$
Simplify each square root: $\sqrt{25} = 5$, $\sqrt{x^2} = x$, $\sqrt{y^4} = y^2$. So we get:
$$5 \times x \times y^2 \times \sqrt{2x} = 5xy^2\sqrt{2x}$$

Problem 25: $\boldsymbol{\sqrt{192a^3b}}$

Step 1: Factor the radicand

Factor $192$ as $64 \times 3$, and $a^3$ as $a^2 \times a$. So:
$$\sqrt{192a^3b} = \sqrt{64 \times 3 \times a^2 \times a \times b}$$

Step 2: Apply the square root property

$$\sqrt{64} \times \sqrt{a^2} \times \sqrt{3ab}$$
Simplify: $\sqrt{64} = 8$, $\sqrt{a^2} = a$. So:
$$8a\sqrt{3ab}$$

Problem 27: $\boldsymbol{\sqrt{112x^3y^4}}$

Step 1: Factor the radicand

Factor $112$ as $16 \times 7$, $x^3$ as $x^2 \times x$, and $y^4$ is a perfect square ($(y^2)^2$). So:
$$\sqrt{112x^3y^4} = \sqrt{16 \times 7 \times x^2 \times x \times y^4}$$

Step 2: Apply the square root property

$$\sqrt{16} \times \sqrt{x^2} \times \sqrt{y^4} \times \sqrt{7x}$$
Simplify each square root: $\sqrt{16} = 4$, $\sqrt{x^2} = x$, $\sqrt{y^4} = y^2$. So:
$$4 \times x \times y^2 \times \sqrt{7x} = 4xy^2\sqrt{7x}$$

Problem 29: $\boldsymbol{\sqrt{216x^3y^3}}$

Step 1: Factor the radicand

Factor $216$ as $36 \times 6$, $x^3$ as $x^2 \times x$, and $y^3$ as $y^2 \times y$. So:
$$\sqrt{216x^3y^3} = \sqrt{36 \times 6 \times x^2 \times x \times y^2 \times y}$$

Step 2: Apply the square root property

$$\sqrt{36} \times \sqrt{x^2} \times \sqrt{y^2} \times \sqrt{6xy}$$
Simplify each square root: $\sqrt{36} = 6$, $\sqrt{x^2} = x$, $\sqrt{y^2} = y$. So:
$$6 \times x \times y \times \sqrt{6xy} = 6xy\sqrt{6xy}$$

Final Answers:

1. $\boldsymbol{3x^2\sqrt{2y}}$
2. $\boldsymbol{5xy^2\sqrt{2x}}$
3. $\boldsymbol{8a\sqrt{3ab}}$
4. $\boldsymbol{4xy^2\sqrt{7x}}$
5. $\boldsymbol{6xy\sqrt{6xy}}$

Answer:

Let's solve each problem one by one. We'll use the property of square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ (where $a \geq 0$ and $b \geq 0$) and simplify the radicand by factoring out perfect squares.

Problem 21: $\boldsymbol{\sqrt{18x^4y}}$

Step 1: Factor the radicand into perfect squares and other factors

Factor $18$ as $9 \times 2$, and $x^4$ is a perfect square ($(x^2)^2$). So we have:
$$\sqrt{18x^4y} = \sqrt{9 \times 2 \times x^4 \times y}$$

Step 2: Apply the square root property

Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get:
$$\sqrt{9} \times \sqrt{x^4} \times \sqrt{2y}$$
Since $\sqrt{9} = 3$ and $\sqrt{x^4} = x^2$, this simplifies to:
$$3x^2\sqrt{2y}$$

Problem 23: $\boldsymbol{\sqrt{50x^3y^4}}$

Step 1: Factor the radicand

Factor $50$ as $25 \times 2$, $x^3$ as $x^2 \times x$, and $y^4$ is a perfect square ($(y^2)^2$). So:
$$\sqrt{50x^3y^4} = \sqrt{25 \times 2 \times x^2 \times x \times y^4}$$

Step 2: Apply the square root property

$$\sqrt{25} \times \sqrt{x^2} \times \sqrt{y^4} \times \sqrt{2x}$$
Simplify each square root: $\sqrt{25} = 5$, $\sqrt{x^2} = x$, $\sqrt{y^4} = y^2$. So we get:
$$5 \times x \times y^2 \times \sqrt{2x} = 5xy^2\sqrt{2x}$$

Problem 25: $\boldsymbol{\sqrt{192a^3b}}$

Step 1: Factor the radicand

Factor $192$ as $64 \times 3$, and $a^3$ as $a^2 \times a$. So:
$$\sqrt{192a^3b} = \sqrt{64 \times 3 \times a^2 \times a \times b}$$

Step 2: Apply the square root property

$$\sqrt{64} \times \sqrt{a^2} \times \sqrt{3ab}$$
Simplify: $\sqrt{64} = 8$, $\sqrt{a^2} = a$. So:
$$8a\sqrt{3ab}$$

Problem 27: $\boldsymbol{\sqrt{112x^3y^4}}$

Step 1: Factor the radicand

Factor $112$ as $16 \times 7$, $x^3$ as $x^2 \times x$, and $y^4$ is a perfect square ($(y^2)^2$). So:
$$\sqrt{112x^3y^4} = \sqrt{16 \times 7 \times x^2 \times x \times y^4}$$

Step 2: Apply the square root property

$$\sqrt{16} \times \sqrt{x^2} \times \sqrt{y^4} \times \sqrt{7x}$$
Simplify each square root: $\sqrt{16} = 4$, $\sqrt{x^2} = x$, $\sqrt{y^4} = y^2$. So:
$$4 \times x \times y^2 \times \sqrt{7x} = 4xy^2\sqrt{7x}$$

Problem 29: $\boldsymbol{\sqrt{216x^3y^3}}$

Step 1: Factor the radicand

Factor $216$ as $36 \times 6$, $x^3$ as $x^2 \times x$, and $y^3$ as $y^2 \times y$. So:
$$\sqrt{216x^3y^3} = \sqrt{36 \times 6 \times x^2 \times x \times y^2 \times y}$$

Step 2: Apply the square root property

$$\sqrt{36} \times \sqrt{x^2} \times \sqrt{y^2} \times \sqrt{6xy}$$
Simplify each square root: $\sqrt{36} = 6$, $\sqrt{x^2} = x$, $\sqrt{y^2} = y$. So:
$$6 \times x \times y \times \sqrt{6xy} = 6xy\sqrt{6xy}$$

Final Answers:

1. $\boldsymbol{3x^2\sqrt{2y}}$
2. $\boldsymbol{5xy^2\sqrt{2x}}$
3. $\boldsymbol{8a\sqrt{3ab}}$
4. $\boldsymbol{4xy^2\sqrt{7x}}$
5. $\boldsymbol{6xy\sqrt{6xy}}$