Simplify square root of 32y^7
The given problem is a mathematical expression simplification task. It involves applying the properties of square roots and exponents. To solve the problem, one would typically break down the expression inside the square root, in this case, 32y^7, into its prime factors and identify pairs of factors. This process would allow for simplifying the square root by taking out factors in pairs (since the square root of a pair of identical factors is just the factor itself) and leaving any unpaired factors within the square root. The goal would be to express the original square root in a simplified form, where all perfect squares have been rooted out, and any remaining factors that do not form a perfect square are left inside the radical.
$\sqrt{32 y^{7}}$
Answer
Solution:
Step 1
Express $32y^{7}$ as the product of a perfect square and a remaining term: $(4y^{3})^{2} \cdot 2y$.
Step 1.1
Extract the perfect square factor from $32$: $\sqrt{16 \cdot 2y^{7}}$.
Step 1.2
Represent $16$ as a square of $4$: $\sqrt{(4^{2}) \cdot 2y^{7}}$.
Step 1.3
Separate $y^{6}$ as a perfect square: $\sqrt{(4^{2}) \cdot 2(y^{6} \cdot y)}$.
Step 1.4
Express $y^{6}$ as $(y^{3})^{2}$: $\sqrt{(4^{2}) \cdot 2((y^{3})^{2} \cdot y)}$.
Step 1.5
Rearrange the terms: $\sqrt{(4^{2}) \cdot (y^{3})^{2} \cdot 2y}$.
Step 1.6
Combine the squares: $\sqrt{((4y^{3})^{2}) \cdot 2y}$.
Step 1.7
Enclose the perfect square and the remaining term in parentheses: $\sqrt{((4y^{3})^{2}) \cdot (2y)}$.
Step 2
Extract the perfect square from the radical: $4y^{3}\sqrt{2y}$.
Knowledge Notes:
The process of simplifying a square root involves several steps:
Prime Factorization: Breaking down a number into its prime factors can help identify perfect squares within the number, which can be taken out of the square root.
Perfect Squares: Recognizing perfect squares is crucial. For example, $16$ is a perfect square because it is $4^{2}$. When a perfect square is under a square root, it can be simplified to its base number.
Properties of Exponents: Understanding that $y^{6}$ can be written as $(y^{3})^{2}$ is an application of the property that $(a^{m})^{n} = a^{mn}$. This is useful when simplifying variables under a square root.
Simplifying Radicals: When a term under a square root is a product of a perfect square and another term, the perfect square can be taken out of the square root, simplifying the expression.
Combining Like Terms: When simplifying expressions, it is often helpful to combine like terms or to rearrange terms to make simplification easier.
By applying these principles, one can systematically simplify square roots of both numerical and algebraic expressions.
