The expression in the image is: [ \sqrt[4]{t} \cdot \sqrt[6]{t^{10}} \cdot t^{-2} ]

Let's simplify this expression step-by-step.

Step 1: Rewrite roots as exponents

We express the radicals as fractional exponents: $\sqrt[4]{t} = t^{\frac{1}{4}}$ $\sqrt[6]{t^{10}} = t^{\frac{10}{6}} = t^{\frac{5}{3}}$

Now, the expression becomes: $t^{\frac{1}{4}} \cdot t^{\frac{5}{3}} \cdot t^{-2}$

Step 2: Combine the exponents

Using the rule of exponents $a^m \cdot a^n = a^{m+n}$, we add the exponents: $\frac{1}{4} + \frac{5}{3} - 2$

To add these fractions, we need a common denominator. The least common denominator (LCD) of 4, 3, and 1 is 12: $\frac{1}{4} = \frac{3}{12}, \quad \frac{5}{3} = \frac{20}{12}, \quad -2 = -\frac{24}{12}$

Now, we add these fractions: $\frac{3}{12} + \frac{20}{12} - \frac{24}{12} = \frac{-1}{12}$

Step 3: Write the simplified expression

The simplified expression is: $t^{-\frac{1}{12}}$

Or, as a radical: $\frac{1}{\sqrt[12]{t}}$

Would you like more details or have any questions?

Follow-up questions

1. How do you convert between radical expressions and fractional exponents?
2. What are the rules for simplifying expressions with negative exponents?
3. How do you find the least common denominator when adding fractions?
4. Can you explain how to simplify an expression like $\sqrt[6]{t^{10}}$ in more detail?
5. How does the product rule for exponents work when combining terms?

Tip:

When working with fractional exponents, it's often easier to convert them into radicals after simplifying the expression.