Simplify the Radical Expression fourth root of 64x^4y^16

Solution:

Step 1

Express $64x^4{y}^{16}$ as $(2x{y}^4{)}^4\cdot 4$.

Step 1.1

Extract $16$ from $64$ to get $\sqrt[4]{16\cdot 4x^4{y}^{16}}$.

Step 1.2

Represent $16$ as $2^4$ to have $\sqrt[4]{2^4\cdot 4x^4{y}^{16}}$.

Step 1.3

Rewrite $y^{16}$ as $(y^4{)}^4$ resulting in $\sqrt[4]{2^4\cdot 4x^4(y^4{)}^4}$.

Step 1.4

Rearrange to place $4$ last: $\sqrt[4]{2^4{x}^4(y^4{)}^4\cdot 4}$.

Step 1.5

Combine $2^4{x}^4(y^4{)}^4$ into $(2x{y}^4{)}^4$: $\sqrt[4]{(2x{y}^4{)}^4\cdot 4}$.

Step 2

Extract terms from under the radical: $2x{y}^4\sqrt[4]{4}$.

Step 3

Eliminate non-negative terms from absolute value: $2y^4|x|\sqrt[4]{4}$.

Step 4

Express $4$ as $2^2$: $2y^4|x|\sqrt[4]{2^2}$.

Step 5

Rewrite $\sqrt[4]{2^2}$ as $\sqrt{\sqrt{2^2}}$: $2y^4|x|\sqrt{\sqrt{2^2}}$.

Step 6

Remove terms from under the square root: $2y^4|x|\sqrt{|2|}$.

Step 7

Recognize that the absolute value of $2$ is $2$: $2y^4|x|\sqrt{2}$.

Knowledge Notes:

To simplify a radical expression, especially one with a higher index like the fourth root, we follow these steps:

1. Factorization: Break down numbers into their prime factors to simplify the radical. For example, $64$ can be factored into $2^6$, and $16$ is $2^4$.

2. Exponent Rules: Use the property that $\sqrt[n]{a^n}=a$ if $a$ is non-negative. This applies to any term with an exponent that is a multiple of the radical's index.

3. Absolute Value: When extracting even roots, we use absolute value to ensure the result is non-negative, since the original expression must represent a real number.

4. Nested Radicals: Simplify nested radicals by recognizing that $\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}$.

5. Simplification: Combine like terms and simplify the expression to its simplest form.

6. Absolute Value with Variables: If the variable under the radical has an even exponent and is being extracted, it is represented with an absolute value to ensure the output is non-negative.

In this problem, we used these principles to simplify the fourth root of $64x^4{y}^{16}$, ultimately extracting terms from under the radical and simplifying the expression to $2y^4|x|\sqrt{2}$.