Simplify square root of 200( square root of 2)
The question is asking for the simplification of a mathematical expression. The expression given is the square root of 200 multiplied by the square root of 2. This is an algebraic problem that involves the manipulation of square roots and their properties. The expected process would involve using properties of radicals to combine and simplify the square roots into a simpler form, typically with the aim of having a rational number outside the radical and the simplest possible expression under the radical.
$\sqrt{200} \left(\right. \sqrt{2} \left.\right)$
Answer
Solution:
Step 1:
Apply the multiplication rule for square roots to combine the terms: $\sqrt{200 \cdot 2}$.
Step 2:
Express the product under the radical as an exponent.
Step 2.1:
Calculate the product of $200$ and $2$: $\sqrt{400}$.
Step 2.2:
Express $400$ as a square of an integer: $\sqrt{(20)^2}$.
Step 2.3:
Extract the square root of the perfect square, assuming we are dealing with positive real numbers: $20$.
Knowledge Notes:
The problem involves simplifying a square root expression that contains another square root. The steps to solve this problem are based on the properties of radicals and exponents. Here are the relevant knowledge points:
Product Rule for Radicals: The square root of a product is equal to the product of the square roots of the individual factors, i.e., $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.
Exponents and Radicals: A number raised to the power of $2$ and then square-rooted yields the original number, i.e., $\sqrt{a^2} = a$ for any positive real number $a$.
Simplifying Square Roots: When simplifying square roots, one looks for perfect squares under the radical to simplify the expression. A perfect square is a number that can be expressed as the square of an integer.
Positive Real Numbers: When pulling terms out from under the radical, it is assumed that we are dealing with positive real numbers to avoid dealing with complex numbers unless specified otherwise.
