Simplify square root of 16^4

Solution:

Step 1:

Compute the fourth power of $16$. We have $16^4 = 65536$. Thus, the expression becomes $\sqrt{65536}$.

Step 2:

Express $65536$ as a square of an integer. It can be written as $(256)^2$. Now, the expression is $\sqrt{(256)^2}$.

Step 3:

Extract the square root of the perfect square. Since we are dealing with positive real numbers, the square root of $(256)^2$ is $256$.

Knowledge Notes:

To simplify the square root of a number raised to a power, such as $\sqrt{16^4}$, we can use the following knowledge points:

1. Exponentiation: Raising a number to a power means multiplying the number by itself as many times as the value of the exponent. For instance, $16^4$ means $16 \times 16 \times 16 \times 16$.

2. Perfect Squares: A perfect square is a number that can be expressed as the square of an integer. For example, $256$ is a perfect square because it can be written as $16^2$.

3. Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a perfect square is always an integer. For example, $\sqrt{256} = 16$ because $16 \times 16 = 256$.

4. Simplification of Radicals: When simplifying the square root of a perfect square, the result is simply the integer whose square is under the radical. For example, $\sqrt{(256)^2} = 256$.

5. Assumption of Positive Real Numbers: When simplifying square roots in the context of real numbers, we typically assume the principal square root, which is the positive root. For example, even though $(-256)^2$ also equals $65536$, when we take the square root, we assume the positive value, $256$.

By applying these concepts, we can simplify the square root of $16^4$ step by step, ultimately finding that it simplifies to $256$.