Question

simplify the following expressions. to be in the most simpli first, then y.
$2x^{\frac{4}{3}}\left(3^{3}x^{9}y^{6}\
ight)^{\frac{4}{3}}$

Explanation:

Step1: Expand the parenthetical term

Apply exponent rule $(a^m b^n c^p)^k = a^{mk} b^{nk} c^{pk}$
$$(3^3 x^9 y^6)^{\frac{4}{3}} = 3^{3 \times \frac{4}{3}} x^{9 \times \frac{4}{3}} y^{6 \times \frac{4}{3}} = 3^4 x^{12} y^8$$

Step2: Calculate constant term

Compute $3^4$
$$3^4 = 81$$

Step3: Multiply with outer terms

Multiply constants and combine like variables using $x^a x^b = x^{a+b}$
$$2x^{\frac{4}{3}} \times 81x^{12}y^8 = (2 \times 81) x^{\frac{4}{3} + 12} y^8$$

Step4: Simplify exponents and constants

Simplify the coefficient and $x$-exponent
$$2 \times 81 = 162, \quad \frac{4}{3} + 12 = \frac{4}{3} + \frac{36}{3} = \frac{40}{3}$$

Answer:

$162x^{\frac{40}{3}}y^8$