Question

s represent positive real numbers).

1. \\(\sqrt{150}\\)
2. \\(\sqrt3{16}\\)
3. \\(\sqrt4{32}\\)
4. \\(\sqrt5{128}\\)
5. \\(\sqrt3{\frac{32}{125}}\\)
6. \\(-\sqrt3{8k^9}\\)
7. \\(-\sqrt3{-125m^9}\\)
8. \\(\sqrt{75y^3}\\)
9. \\(\sqrt3{8z^9r^{12}}\\)
10. \\(\sqrt4{16a^8b^{12}}\\)

Explanation:

Step1: Simplify $\sqrt{150}$

Factor 150 into perfect square: $150=25\times6$
$\sqrt{150}=\sqrt{25\times6}=\sqrt{25}\times\sqrt{6}=5\sqrt{6}$

Step2: Simplify $\sqrt[3]{16}$

Factor 16 into perfect cube: $16=8\times2$
$\sqrt[3]{16}=\sqrt[3]{8\times2}=\sqrt[3]{8}\times\sqrt[3]{2}=2\sqrt[3]{2}$

Step3: Simplify $\sqrt[4]{32}$

Factor 32 into perfect 4th power: $32=16\times2$
$\sqrt[4]{32}=\sqrt[4]{16\times2}=\sqrt[4]{16}\times\sqrt[4]{2}=2\sqrt[4]{2}$

Step4: Simplify $\sqrt[5]{128}$

Factor 128 into perfect 5th power: $128=32\times4$
$\sqrt[5]{128}=\sqrt[5]{32\times4}=\sqrt[5]{32}\times\sqrt[5]{4}=2\sqrt[5]{4}$

Step5: Simplify $\sqrt[3]{\frac{32}{125}}$

Split root, factor numerator: $\sqrt[3]{\frac{32}{125}}=\frac{\sqrt[3]{32}}{\sqrt[3]{125}}=\frac{\sqrt[3]{8\times4}}{5}=\frac{2\sqrt[3]{4}}{5}$

Step6: Simplify $-\sqrt[3]{8k^9}$

Factor term inside root: $8k^9=8(k^3)^3$
$-\sqrt[3]{8k^9}=-\sqrt[3]{8}\times\sqrt[3]{(k^3)^3}=-2k^3$

Step7: Simplify $-\sqrt[3]{-125m^9}$ (assuming missing term is $m^9$)

Negative inside cube root: $-\sqrt[3]{-125m^9}=-(-\sqrt[3]{125m^9})=\sqrt[3]{125}\times\sqrt[3]{(m^3)^3}=5m^3$

Step8: Simplify $\sqrt{75y^3}$

Factor into perfect square: $75y^3=25y^2\times3y$
$\sqrt{75y^3}=\sqrt{25y^2}\times\sqrt{3y}=5y\sqrt{3y}$

Step9: Simplify $\sqrt[3]{8z^9r^{12}}$

Factor into perfect cube: $8z^9r^{12}=8(z^3)^3(r^4)^3$
$\sqrt[3]{8z^9r^{12}}=\sqrt[3]{8}\times\sqrt[3]{(z^3)^3}\times\sqrt[3]{(r^4)^3}=2z^3r^4$

Step10: Simplify $\sqrt[4]{16a^8b^{12}}$

Factor into perfect 4th power: $16a^8b^{12}=16(a^2)^4(b^3)^4$
$\sqrt[4]{16a^8b^{12}}=\sqrt[4]{16}\times\sqrt[4]{(a^2)^4}\times\sqrt[4]{(b^3)^4}=2a^2b^3$

Answer:

1. $5\sqrt{6}$
2. $2\sqrt[3]{2}$
3. $2\sqrt[4]{2}$
4. $2\sqrt[5]{4}$
5. $\frac{2\sqrt[3]{4}}{5}$
6. $-2k^3$
7. $5m^3$ (assuming the full term is $-125m^9$)
8. $5y\sqrt{3y}$
9. $2z^3r^4$
10. $2a^2b^3$