To simplify the expression $w^{1/8} x w^{4/5}$, follow these steps:

1. The two terms with the same base $w$ (i.e., $w^{1/8}$ and $w^{4/5}$) can be combined by adding their exponents using the exponent rule $a^m \cdot a^n = a^{m+n}$.

$w^{1/8} \cdot w^{4/5} = w^{(1/8 + 4/5)}$

2. Find a common denominator to add the fractions. The least common denominator of 8 and 5 is 40.

$\frac{1}{8} = \frac{5}{40}, \quad \frac{4}{5} = \frac{32}{40}$

3. Add the fractions:

$\frac{5}{40} + \frac{32}{40} = \frac{37}{40}$

4. So, the simplified expression is:

$w^{37/40} x$

Thus, the simplified form of the expression is:

$w^{37/40} x$

Would you like further clarification or have any questions?

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Here are 5 related questions:

1. How do you simplify expressions involving negative exponents?
2. Can you simplify $w^{3/4} \cdot w^{1/2}$?
3. What happens if there are coefficients with the same base in an exponent expression?
4. How do you convert fractional exponents into radicals?
5. How do you divide powers with the same base?

Tip: When adding or subtracting fractions, always find a common denominator for accurate results.