Simplify 2xy fourth root of y*(3 fourth root of 3y fourth root of y^3)
The provided question pertains to algebra and involves the simplification of an expression containing variables, coefficients, and radicals (specifically fourth roots). The expression provided is a product of multiple terms including a variable x, a variable y, and fourth roots of various expressions that involve the variable y and the constant 3. The question is asking for a simplification of this expression by combining like terms, applying the properties of exponents and radicals (like multiplying roots with the same index), and reducing the expression to its simplest form.
$2 x y \sqrt[4]{y} \cdot \left(\right. 3 \sqrt[4]{3 y} \sqrt[4]{y^{3}} \left.\right)$
Answer
Solution:
Step:1
Execute the multiplication of $3 \sqrt[4]{3y} \sqrt[4]{y^3}$.
Step:1.1
Apply the radical product rule to $2xy \sqrt[4]{y} \cdot (3 \sqrt[4]{3y^3})$.
Step:1.2
Combine the powers of $y$ by summing their exponents.
Step:1.2.1
Rearrange to place $y$ adjacent to $y^3$: $2xy \sqrt[4]{y} \cdot (3 \sqrt[4]{3 \cdot y \cdot y^3})$.
Step:1.2.2
Multiply $y$ and $y^3$ together.
Step:1.2.2.1
Express $y$ as $y^1$: $2xy \sqrt[4]{y} \cdot (3 \sqrt[4]{y^1 \cdot y^3 \cdot 3})$.
Step:1.2.2.2
Combine the exponents using the power rule $a^m a^n = a^{m+n}$: $2xy \sqrt[4]{y} \cdot (3 \sqrt[4]{y^{1+3} \cdot 3})$.
Step:1.2.3
Add the exponents $1$ and $3$: $2xy \sqrt[4]{y} \cdot (3 \sqrt[4]{y^4 \cdot 3})$.
Step:2
Extract terms from under the radical, recognizing that $y^4$ is a perfect fourth power: $2xy \sqrt[4]{y} \cdot (3y \sqrt[4]{3})$.
Step:3
Combine the $y$ terms by adding their exponents.
Step:3.1
Group the $y$ terms together: $2x(y \cdot y) \sqrt[4]{y} \cdot (3 \sqrt[4]{3})$.
Step:3.2
Multiply the $y$ terms: $2xy^2 \sqrt[4]{y} \cdot (3 \sqrt[4]{3})$.
Step:4
Multiply $2xy^2 \sqrt[4]{y}$ by $(3 \sqrt[4]{3})$.
Step:4.1
Multiply $3$ by $2$: $6xy^2 \sqrt[4]{y} \sqrt[4]{3}$.
Step:4.2
Apply the radical product rule: $6xy^2 \sqrt[4]{3y}$.
Knowledge Notes:
The problem involves simplifying an expression that contains radicals and algebraic terms. The key knowledge points and rules applied in the solution are:
Product Rule for Radicals: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$, which allows us to multiply two radicals with the same index (the root) together by multiplying the radicands (the numbers or expressions inside the radical).
Exponent Rules: When multiplying like bases, we add the exponents: $a^m \cdot a^n = a^{m+n}$. This is used to combine terms with the same base during multiplication.
Simplifying Radicals: When a radicand is a perfect power of the index of the radical, it can be simplified. For example, $\sqrt[4]{y^4} = y$ because $y^4$ is a perfect fourth power.
Combining Like Terms: Terms that are the same can be combined by adding or subtracting their coefficients.
Multiplication of Coefficients: When multiplying terms, we multiply the coefficients (the numerical parts) and apply the exponent rules to the variables.
The solution applies these rules step by step to simplify the given algebraic expression.
