3 (in the nook) of the radical of 8 raised to the second power according to my book equals 2^2 which equals for. But how and why does this equal 2^2? Can you explain the process please? I don't get how that radical equals four! Thanks for your help!

your description of 3 in the nook means the cube root.

(cubroot 8)^2
2^2
4

Certainly! I'd be happy to explain the process to you.

To better understand how the radical of 8 raised to the second power equals 2^2, let's break it down step by step:

Step 1: Simplify the expression inside the radical.
We have the radical of 8 raised to the second power, which means we have (sqrt(8))^2.

Step 2: Simplify the expression inside the parentheses.
The square root of 8 can also be written as sqrt(4 * 2), where 4 is a perfect square.

Step 3: Apply the properties of radicals.
Using the property of radicals known as the product property, we can write sqrt(4 * 2) as sqrt(4) * sqrt(2). The square root of 4 is 2, so we are left with 2 * sqrt(2).

Step 4: Simplify the expression.
Multiplying 2 by sqrt(2), we get 2*sqrt(2).

So ultimately, the radical of 8 raised to the second power simplifies to 2*sqrt(2), not 2^2.

However, if there was a typo in your book and the radical of 8 raised to the second power should actually be written as (sqrt(8))^2, then it can be simplified differently as follows:

Step 1: Simplify the expression inside the parentheses.
The square root of 8 can also be written as sqrt(4 * 2).

Step 2: Apply the properties of radicals.
Using the product property of radicals, sqrt(4 * 2) can be written as sqrt(4) * sqrt(2), which equals 2 * sqrt(2).

Step 3: Apply the exponent.
Now that we have (2 * sqrt(2))^2, we can square the expression. Squaring a product is the same as squaring each individual factor, so we have (2^2) * (sqrt(2))^2.

Step 4: Simplify the expression.
Since 2^2 equals 4 and (sqrt(2))^2 also equals 2, we end up with 4 * 2, which equals 8.

Therefore, if the expression was written as (sqrt(8))^2, then the answer is 8, not 4.