Simplify square root of 40x^2y^3z^8
This question is asking for the simplification of a radical expression, specifically the square root of the given algebraic expression, which includes numerical and variable components (40x^2y^3z^8). The task involves using properties of square roots and exponents to break down the expression into its simplest form, where no perfect squares remain under the radical sign if possible, and the exponents are simplified according to the rules of roots and powers.
$\sqrt{40 x^{2} y^{3} z^{8}}$
Answer
Solution:
Step 1: Decompose the expression
Express $40x^2y^3z^8$ as a product of squares and other factors to simplify the square root.
Step 1.1: Extract the square factor from 40
Identify the perfect square within the number 40: $\sqrt{4 \cdot 10 x^2 y^3 z^8}$.
Step 1.2: Represent 4 as a square
Express 4 as the square of 2: $\sqrt{2^2 \cdot 10 x^2 y^3 z^8}$.
Step 1.3: Separate the square of y
Factor out the square of y from $y^3$: $\sqrt{2^2 \cdot 10 x^2 (y^2 y) z^8}$.
Step 1.4: Rewrite $z^8$ as a square
Recognize $z^8$ as a square of $z^4$: $\sqrt{2^2 \cdot 10 x^2 (y^2 y) (z^4)^2}$.
Step 1.5: Rearrange y
Position the single y outside the perfect squares: $\sqrt{2^2 \cdot 10 x^2 y^2 (z^4)^2 y}$.
Step 1.6: Move the 10
Place the number 10 outside the perfect square terms: $\sqrt{2^2 x^2 y^2 (z^4)^2 \cdot 10 y}$.
Step 1.7: Combine the squares
Combine the squares into a single term: $\sqrt{(2xy z^4)^2 \cdot 10 y}$.
Step 1.8: Enclose in parentheses
Encapsulate the expression properly: $\sqrt{(2xy z^4)^2 \cdot (10 y)}$.
Step 2: Simplify the radical
Extract the square terms from under the square root to simplify the expression: $2xyz^4 \sqrt{10 y}$.
Knowledge Notes:
To simplify a square root involving variables and coefficients, we follow these steps:
Factorization: Break down the coefficient into its prime factors and group them into pairs of equal factors. For variables, group them into pairs of equal exponents.
Perfect Squares: Identify perfect square factors within the expression. A perfect square is a number or a variable raised to an even power, which can be simplified when under a square root.
Square Root Properties: Use the property that the square root of a product is the product of the square roots, and the square root of a square is the original number or variable.
Extraction: Pull out the square terms from under the square root. When a perfect square is under a square root, it can be taken out of the radical, leaving no radical behind for that part of the expression.
Simplification: After extracting the squares, simplify the remaining expression under the square root if possible.
In this problem, we used these principles to simplify $\sqrt{40x^2y^3z^8}$ by identifying and extracting the perfect squares and then simplifying the remaining expression under the square root.
