Simplify -4 fourth root of 81

Solution:

Simplification Process:

Step 1: Express the number 81 as a power of 3, which is $3^4$. Thus, the expression becomes $-4\sqrt[4]{3^4}$.

Step 2: Extract the fourth root of $3^4$ from under the radical, which is 3, as we are considering only the principal root for real numbers. The expression now is $-4\cdot 3$.

Step 3: Perform the multiplication of $-4$ and $3$ to obtain the final simplified result, which is $-12$.

Knowledge Notes:

To simplify an expression like $-4\sqrt[4]{81}$, we can use the following knowledge points:

1. Radicals and Exponents: The fourth root of a number is the same as raising that number to the power of $\frac{1}{4}$. If the number under the radical is a perfect fourth power, it can be simplified by taking the fourth root directly.

2. Rewriting Numbers: Numbers like 81 can be rewritten as a power of their prime factors. In this case, 81 is $3^4$.

3. Simplifying Radicals: When the exponent of the number under the radical matches the index of the radical (in this case, both are 4), the radical simplifies to the base of the exponent. Therefore, $\sqrt[4]{3^4}=3$.

4. Multiplication of Integers: When multiplying integers, if both have the same sign, the result is positive. If they have different signs, the result is negative. Here, $-4$ (negative) multiplied by $3$ (positive) gives $-12$ (negative).

5. Principal Root: When dealing with even roots (like square roots or fourth roots), the principal root is assumed, which is the positive root. However, since we have a negative sign in front of the radical, the result will be negative after multiplication.

By combining these principles, we can simplify the given expression step by step, ensuring that each operation follows the rules of arithmetic and algebra.