Question

3. Simplify and express the answer with positive exponent. ​

Answer 1

Question :-

Simplify and express the answer with positive exponent -

$\sf \Big[ \sqrt[3]{x^4 y} \times \dfrac{1}{\sqrt[3]{xy^7}}\Big]^{-4}$

⠀

Solution :-

$\implies\sf \Big[ \sqrt[3]{x^4 y} \times \dfrac{1}{\sqrt[3]{xy^7}}\Big]^{-4}$

⠀

• $\sf \sqrt a = (a)^{\dfrac{1}{n}}$

⠀

$\implies\sf \Big[x^{4 \times \frac{1}{3}} \times y^{\frac{1}{3}} \times \dfrac{1}{x^\frac{1}{3} \times y^{7 \times \frac{1}{3}} }\Big]^{-4}$

⠀

$\implies\sf \Big[x^{\frac{4}{3}} \times y^{\frac{1}{3}} \times \dfrac{1}{x^\frac{1}{3} \times y^{\frac{7}{3}} }\Big]^{-4}$

⠀

• $\sf \dfrac{1}{a^n} = a^{-n}$

⠀

$\implies\sf [x^{\frac{4}{3}} \times y^{\frac{1}{3}} \times x^{-\frac{1}{3}} \times y^{-\frac{7}{3}}]^{-4}$

⠀

• $\sf a^m \times a^n = a^{m + n}$

⠀

$\implies\sf [x^{\frac{4}{3} + ( - \frac{1}{3})} \times y^{\frac{1}{3} + ( - \frac{7}{3})}]^{-4}$

⠀

$\implies\sf [x^{\frac{4}{3} - \frac{1}{3}} \times y^{\frac{1}{3} - \frac{7}{3}}]^{-4}$

⠀

$\implies\sf [x^{\frac{4-1}{3}} \times y^\frac{1 - 7}{3}]^{-4}$

⠀

$\implies\sf [x^{\frac{3}{3}} \times y^{\frac{-6}{3}}]^{-4}$

⠀

$\implies\sf [x^1 \times y^{-2}]^{-4}$

⠀

$\implies\sf (x^1)^{-4} \times (y^{-2})^{-4}$

⠀

• $\sf (a^m)^n = a^{mn}$

⠀

$\implies\sf x^{-4} \times y^{(-2) \times (-4)}$

⠀

$\implies\sf x^{-4} \times y^{8}$

⠀

$\implies\boxed{\sf \dfrac{y^8}{x^4}}$