Simplify the Radical Expression fourth root of 64x^4y^16
The question prompts you to simplify the given radical expression, which involves finding the fourth root of a numerical and algebraic expression: 64x^4y^16. You're expected to apply the properties of exponents and roots to express the content inside the radical in its simplest form, which would no longer involve a radical sign if possible. Essentially, you need to find the smallest expression that, when raised to the fourth power, equals the original expression under the radical.
$\sqrt[4]{64 x^{4} y^{16}}$
Answer
Solution:
Step 1
Express $64x^4y^{16}$ as $(2xy^4)^4 \cdot 4$.
Step 1.1
Extract $16$ from $64$ to get $\sqrt[4]{16 \cdot 4 x^4 y^{16}}$.
Step 1.2
Represent $16$ as $2^4$ to have $\sqrt[4]{2^4 \cdot 4 x^4 y^{16}}$.
Step 1.3
Rewrite $y^{16}$ as $(y^4)^4$ resulting in $\sqrt[4]{2^4 \cdot 4 x^4 (y^4)^4}$.
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 $(2xy^4)^4$: $\sqrt[4]{(2xy^4)^4 \cdot 4}$.
Step 2
Extract terms from under the radical: $2xy^4 \sqrt[4]{4}$.
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:
Factorization: Break down numbers into their prime factors to simplify the radical. For example, $64$ can be factored into $2^6$, and $16$ is $2^4$.
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.
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.
Nested Radicals: Simplify nested radicals by recognizing that $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a}$.
Simplification: Combine like terms and simplify the expression to its simplest form.
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 $64x^4y^{16}$, ultimately extracting terms from under the radical and simplifying the expression to $2y^4 |x| \sqrt{2}$.
