Question

radicals
simplify.

1. $\sqrt{24x^4}$
2. $\sqrt{448n^2}$
3. $\sqrt{196m^3}$
4. $\sqrt{36x^2y}$
5. $\sqrt{150x^3y}$
6. $-5\sqrt{162m^4n^4p}$
7. $-2\sqrt{18p^8q^2r^2}$

Explanation:

Step1: Factor out perfect squares

$\sqrt{24x^4} = \sqrt{4x^4 \cdot 6}$

Step2: Simplify perfect square roots

$\sqrt{4x^4} \cdot \sqrt{6} = 2x^2\sqrt{6}$

---

Step1: Factor out perfect squares

$\sqrt{448n^2} = \sqrt{64n^2 \cdot 7}$

Step2: Simplify perfect square roots

$\sqrt{64n^2} \cdot \sqrt{7} = 8|n|\sqrt{7}$

---

Step1: Factor out perfect squares

$\sqrt{196m^3} = \sqrt{196m^2 \cdot m}$

Step2: Simplify perfect square roots

$\sqrt{196m^2} \cdot \sqrt{m} = 14|m|\sqrt{m}$

---

Step1: Factor out perfect squares

$\sqrt{36x^2y} = \sqrt{36x^2 \cdot y}$

Step2: Simplify perfect square roots

$\sqrt{36x^2} \cdot \sqrt{y} = 6|x|\sqrt{y}$

---

Step1: Factor out perfect squares

$\sqrt{150x^3y} = \sqrt{25x^2 \cdot 6xy}$

Step2: Simplify perfect square roots

$\sqrt{25x^2} \cdot \sqrt{6xy} = 5|x|\sqrt{6xy}$

---

Step1: Factor out perfect squares

$-5\sqrt{162m^4n^4p} = -5\sqrt{81m^4n^4 \cdot 2p}$

Step2: Simplify perfect square roots

$-5 \cdot \sqrt{81m^4n^4} \cdot \sqrt{2p} = -5 \cdot 9m^2n^2\sqrt{2p}$

Step3: Multiply coefficients

$-45m^2n^2\sqrt{2p}$

---

Step1: Factor out perfect squares

$-2\sqrt{18p^4q^2r^2} = -2\sqrt{9p^4q^2r^2 \cdot 2}$

Step2: Simplify perfect square roots

$-2 \cdot \sqrt{9p^4q^2r^2} \cdot \sqrt{2} = -2 \cdot 3p^2|q||r|\sqrt{2}$

Step3: Multiply coefficients

$-6p^2|q||r|\sqrt{2}$

Answer:

1. $2x^2\sqrt{6}$
2. $8|n|\sqrt{7}$
3. $14|m|\sqrt{m}$
4. $6|x|\sqrt{y}$
5. $5|x|\sqrt{6xy}$
6. $-45m^2n^2\sqrt{2p}$
7. $-6p^2|q||r|\sqrt{2}$