Question

① $sqrt5{y^{5}}$ ② $sqrt{x^{4}y^{12}}$ ③ $sqrt3{a^{6}b^{3}}$ ④ $16^{\frac{1}{5}}cdot 16^{\frac{4}{5}}$ ⑤ $\frac{27^{\frac{1}{2}}}{27^{\frac{1}{6}}}$ ⑥ $sqrt{25y^{4}}$

Explanation:

Step1: Simplify fifth root of $y^5$

$\sqrt[5]{y^5} = y$

Step2: Simplify square root of $x^4y^{12}$

$\sqrt{x^4y^{12}} = (x^4y^{12})^{\frac{1}{2}} = x^{2}y^{6}$

Step3: Simplify cube root of $a^6b^3$

$\sqrt[3]{a^6b^3} = (a^6b^3)^{\frac{1}{3}} = a^{2}b$

Step4: Add exponents for same base

$16^{\frac{1}{5}} \cdot 16^{\frac{4}{5}} = 16^{\frac{1}{5}+\frac{4}{5}} = 16^{1} = 16$

Step5: Subtract exponents for same base

$\frac{27^{\frac{1}{2}}}{27^{\frac{1}{6}}} = 27^{\frac{1}{2}-\frac{1}{6}} = 27^{\frac{1}{3}} = 3$

Step6: Simplify square root of $25y^4$

$\sqrt{25y^4} = (25y^4)^{\frac{1}{2}} = 5y^{2}$

Answer:

1. $y$
2. $x^2y^6$
3. $a^2b$
4. $16$
5. $3$
6. $5y^2$