Simplify the expression $(8y^3x^{27}x^3)^{\frac{1}{3}}$.

1. Problem Description

Simplify the expression $(8y^3x^{27}x^3)^{\frac{1}{3}}$.

2. Solution Steps

First, we simplify the expression inside the parentheses by combining the $x$ terms:

$x^{27}x^3 = x^{27+3} = x^{30}$.

So the expression becomes $(8y^3x^{30})^{\frac{1}{3}}$.

Next, we apply the exponent $\frac{1}{3}$ to each term in the parentheses:

$(ab)^n = a^n b^n$

$(8y^3x^{30})^{\frac{1}{3}} = 8^{\frac{1}{3}} (y^3)^{\frac{1}{3}} (x^{30})^{\frac{1}{3}}$.

We know that $8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2^{3*\frac{1}{3}} = 2^1 = 2$.

Using the rule $(a^m)^n = a^{m*n}$, we have:

$(y^3)^{\frac{1}{3}} = y^{3*\frac{1}{3}} = y^1 = y$.

$(x^{30})^{\frac{1}{3}} = x^{30*\frac{1}{3}} = x^{10}$.

Therefore, $(8y^3x^{30})^{\frac{1}{3}} = 2yx^{10}$.

3. Final Answer

$2yx^{10}$