## Understanding (a^2b)^4

In mathematics, the expression (a^2b)^4 represents the **fourth power of the product of a squared and b**. Let's break down this expression to understand how it simplifies:

### Understanding the Components

**a^2:**This represents 'a' multiplied by itself twice (a * a).**b:**This represents the variable 'b'.**( )^4:**This indicates that the entire expression within the parentheses is raised to the power of 4.

### Simplifying the Expression

To simplify (a^2b)^4, we apply the **power of a product rule**. This rule states that raising a product to a power is the same as raising each factor to that power.

In our case:

(a^2b)^4 = (a^2)^4 * (b)^4

Now we use another rule: **power of a power rule**. This rule states that raising a power to another power is the same as multiplying the exponents.

Applying this rule:

(a^2)^4 * (b)^4 = a^(2*4) * b^4

Simplifying further:

a^(2*4) * b^4 = **a^8 * b^4**

### Conclusion

Therefore, (a^2b)^4 simplifies to **a^8 * b^4**. This means that the expression represents 'a' multiplied by itself eight times, and then multiplied by 'b' multiplied by itself four times. This understanding is crucial for simplifying and manipulating algebraic expressions in various mathematical contexts.