The expression in the image you provided can be simplified step by step. Let's break it down.

We have:

$\sqrt[7]{\frac{x^{14} \cdot y^{21} \cdot z^{35}}{y^{14} \cdot z^7}}$

Step 1: Simplify the numerator and denominator

First, simplify the fraction inside the radical.

Numerator: $x^{14} \cdot y^{21} \cdot z^{35}$

Denominator: $y^{14} \cdot z^7$

Now, cancel out the common powers of $y$ and $z$:

$\frac{y^{21}}{y^{14}} = y^{21 - 14} = y^7$ $\frac{z^{35}}{z^7} = z^{35 - 7} = z^{28}$

The fraction now becomes:

$\frac{x^{14} \cdot y^7 \cdot z^{28}}{1}$

Step 2: Apply the 7th root to each term

Now we apply the 7th root to the simplified expression:

[ \sqrt[7]{x^{14} \cdot y^7 \cdot z^{28}} = \sqrt[7]{x^{14}} \cdot \sqrt[7]{y^7} \cdot \sqrt[7]{z^{28}} ]

Now simplify each term:

$\sqrt[7]{x^{14}} = x^{14/7} = x^2$ $\sqrt[7]{y^7} = y^{7/7} = y$ $\sqrt[7]{z^{28}} = z^{28/7} = z^4$

Step 3: Final simplified expression

Putting everything together, the simplified form of the expression is:

$x^2 \cdot y \cdot z^4$

So, the expression simplifies to $x^2 y z^4$.

Do you need further details or have any questions? Here are some related topics you might be interested in:

1. How to simplify other types of fractional exponents
2. The properties of radicals in algebra
3. How to simplify expressions involving roots and exponents
4. How to handle negative exponents in radical expressions
5. Real-world applications of exponents and roots in mathematics

Tip: When simplifying expressions with exponents and roots, it's helpful to separate the variables and deal with them individually for easier manipulation.