Simplify square root of 50x^5y^8
This question involves simplifying a radical expression which contains both numerical and algebraic components. You are required to simplify the square root of the product of 50, x to the power of 5, and y to the power of 8. This entails finding the principal square root and expressing the result in its simplest radical form, where all perfect square factors have been removed from under the radical sign. This can include breaking down the number 50 into its prime factors, simplifying the variable expressions by applying the properties of exponents, and possibly separating the result into a product of simpler square roots.
$\sqrt{50 x^{5} y^{8}}$
Answer
Solution:
Step 1: Express the given expression in a factorizable form
Rewrite the expression $50x^5y^8$ as $(5x^2y^4)^2 \cdot 2x$.
Step 1.1: Extract the perfect square factor from 50
Factor out the perfect square 25 from 50: $\sqrt{25 \cdot 2x^5y^8}$.
Step 1.2: Represent 25 as a square of 5
Express 25 as $5^2$: $\sqrt{5^2 \cdot 2x^5y^8}$.
Step 1.3: Separate the perfect square $x^4$ from $x^5$
Factor $x^4$ out of $x^5$: $\sqrt{5^2 \cdot 2(x^4 \cdot x)y^8}$.
Step 1.4: Rewrite $x^4$ as $(x^2)^2$
Express $x^4$ as $(x^2)^2$: $\sqrt{5^2 \cdot 2((x^2)^2 \cdot x)y^8}$.
Step 1.5: Rewrite $y^8$ as $(y^4)^2$
Express $y^8$ as $(y^4)^2$: $\sqrt{5^2 \cdot 2((x^2)^2 \cdot x)(y^4)^2}$.
Step 1.6: Rearrange the expression to isolate $x$
Rearrange to isolate the single $x$: $\sqrt{5^2 \cdot 2((x^2)^2)(y^4)^2 \cdot x}$.
Step 1.7: Rearrange the expression to isolate 2
Rearrange to isolate the factor 2: $\sqrt{(5^2((x^2)^2)(y^4)^2) \cdot 2x}$.
Step 1.8: Combine the squares into a single term
Combine the squared terms: $\sqrt{(5x^2y^4)^2 \cdot 2x}$.
Step 1.9: Enclose the terms properly with parentheses
Ensure proper use of parentheses: $\sqrt{((5x^2y^4)^2 \cdot (2x))}$.
Step 2: Simplify the square root expression
Extract terms from under the square root: $5x^2y^4\sqrt{2x}$.
Knowledge Notes:
To simplify a square root expression involving variables and constants, we follow these steps:
Factorization: Break down the expression under the square root into factors, especially looking for perfect squares and variables with even exponents, as these can be simplified outside the square root.
Rewriting as Squares: Express constants that are perfect squares (like 25) as the square of their square roots (like $5^2$). Do the same for variables with even exponents (like $x^4$ as $(x^2)^2$).
Extraction of Perfect Squares: Any term under the square root that is a perfect square can be taken out of the square root by writing it without the square root and halving its exponent.
Rearrangement: Sometimes, rearranging the terms under the square root can make the simplification process more straightforward.
Simplification: After extracting perfect squares, simplify the expression by multiplying the terms outside the square root and leaving the rest inside the square root.
In this problem, we used these principles to simplify the square root of $50x^5y^8$ by identifying and extracting perfect squares such as $5^2$, $(x^2)^2$, and $(y^4)^2$. The remaining terms that are not perfect squares stay under the square root.
