Simplify square root of 2( square root of 11- square root of 7)
This mathematical expression involves nested square roots and multiplication. The problem is asking you to simplify the expression by performing the operations inside the parentheses first and then applying the distributive property of multiplication over subtraction to combine the terms inside the square root with the square root of 2. This process will result in an expression that has potentially fewer square roots and is in its simplest algebraic form.
$\sqrt{2} \left(\right. \sqrt{11} - \sqrt{7} \left.\right)$
Answer
Solution:
Step:1
Utilize the distributive property to expand the expression: $\sqrt{2}(\sqrt{11} - \sqrt{7})$ becomes $\sqrt{2}\sqrt{11} - \sqrt{2}\sqrt{7}$.
Step:2
Simplify the terms by applying the multiplication rule for square roots: $\sqrt{2}\sqrt{11}$ becomes $\sqrt{2 \cdot 11}$ and $\sqrt{2}\sqrt{7}$ becomes $\sqrt{2 \cdot 7}$.
Step:3
Perform the multiplication inside the square roots.
Step:3.1
Calculate $\sqrt{2 \cdot 11}$ to get $\sqrt{22}$.
Step:3.2
Calculate $\sqrt{2 \cdot 7}$ to get $\sqrt{14}$.
Step:4
Combine the results of the multiplications: $\sqrt{22} - \sqrt{14}$.
Step:5
Present the final expression in its exact and decimal forms:
Exact Form: $\sqrt{22} - \sqrt{14}$ Decimal Form: Approximately $0.94875837 \ldots$
Knowledge Notes:
The problem involves simplifying a radical expression using properties of square roots and the distributive property. Here are the relevant knowledge points:
Distributive Property: This property states that for any numbers $a$, $b$, and $c$, the expression $a(b + c)$ is equal to $ab + ac$. It allows us to expand expressions where a term is being multiplied by a sum or difference.
Product Rule for Radicals: The product rule for radicals states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, where $a$ and $b$ are non-negative numbers. This rule is used to combine and simplify square roots.
Simplifying Square Roots: To simplify a square root, one needs to find the prime factorization of the number under the radical and then apply the product rule for radicals if possible.
Exact vs. Decimal Form: The exact form of a radical expression is the form that includes square roots and cannot be simplified further without approximations. The decimal form is a numerical approximation of the exact form, often rounded to a certain number of decimal places.
Latex Formatting: In the solution, we use Latex to render mathematical expressions, such as $\sqrt{2 \cdot 11}$ and $\sqrt{2 \cdot 7}$, to make them clear and visually consistent with mathematical notation.
