Multiply fourth root of 8* fourth root of 8

Solution:

Step 1:

Apply the product property of radicals to combine them. $\sqrt[4]{8\times 8}$

Step 2:

Calculate the product inside the radical. $\sqrt[4]{64}$

Step 3:

Express 64 as a product of its prime factors.

Step 3.1:

Extract the factor of 16 from 64. $\sqrt[4]{16\cdot 4}$

Step 3.2:

Represent 16 as a power of 2. $\sqrt[4]{2^4\cdot 4}$

Step 4:

Simplify the radical by removing perfect fourth powers. $2\sqrt[4]{4}$

Step 5:

Express 4 as a power of 2. $2\sqrt[4]{2^2}$

Step 6:

Convert the fourth root of a square to the square root of a square root. $2\sqrt{\sqrt{2^2}}$

Step 7:

Simplify the nested radicals, assuming all numbers are positive. $2\sqrt{2}$

Step 8:

Present the final answer in its exact and decimal forms.

Exact Form: $2\sqrt{2}$ Decimal Form: $2.82842712\dots$

Knowledge Notes:

The problem involves multiplying two identical fourth roots, which is a specific case of the more general product rule for radicals. The product rule states that $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{ab}$, where $n$ is the index of the radical and $a$ and $b$ are the radicands.

In this case, we are dealing with the fourth root, which means we are looking for a number that, when raised to the fourth power, gives the radicand. When the radicand is a perfect fourth power, the radical simplifies neatly.

The process of simplifying involves several steps:

1. Combine the radicals using the product rule.

2. Multiply the numbers inside the radical.

3. Factor the result into a product of a perfect fourth power and another factor.

4. Simplify by taking the fourth root of the perfect fourth power, which can be done by raising the base to the power of one-fourth.

5. If the remaining factor inside the radical is a perfect square, it can be further simplified by recognizing that the fourth root of a square is the square root of the square root.

6. Finally, the expression is simplified to its most reduced form.

It's important to note that when dealing with even roots, we typically assume that we are working with the principal (non-negative) root. This is why the final answer is given as $2\sqrt{2}$, assuming that all numbers are positive.