Simplify square root of 3( square root of 12+ square root of 3)
The given problem is asking to carry out the simplification of a mathematical expression that involves square roots. Specifically, it is asking to simplify the product of the square root of 3 and a sum that includes another two square roots: the square root of 12 and the square root of 3. Simplifying this expression will likely involve using properties of square roots, such as the product rule for square roots and the ability to reduce square roots of perfect squares.
$\sqrt{3 \left(\right. \sqrt{12 + \sqrt{3}} \left.\right)}$
Answer
Solution:
Step 1:
Compute the product of $3$ and the expression within the square root, $\sqrt{12 + \sqrt{3}}$, to obtain $\sqrt{3 \cdot \sqrt{12 + \sqrt{3}}}$.
Step 2:
The expression can be represented in different formats. For the exact representation, we have $\sqrt{3 \cdot \sqrt{12 + \sqrt{3}}}$. If we convert it to a decimal approximation, the result is $3.33422173\ldots$.
Knowledge Notes:
The problem involves simplifying a nested radical expression. Here are the relevant knowledge points:
Nested Radicals: A nested radical is a radical expression (an expression with a root, such as a square root) that contains another radical expression. Simplifying nested radicals often involves recognizing patterns or using conjugates.
Simplifying Square Roots: To simplify a square root, you look for perfect square factors of the number under the root. For example, $\sqrt{12}$ can be simplified to $\sqrt{4 \cdot 3}$, which is $2\sqrt{3}$.
Multiplication of Radicals: When multiplying radicals, you can use the property $\sqrt{a}\sqrt{b} = \sqrt{ab}$ to combine them into a single radical.
Exact vs. Decimal Form: The exact form of a radical expression is left in terms of square roots, without any approximation. The decimal form is a numerical approximation of the radical expression, often rounded to a certain number of decimal places.
LaTeX Formatting: When writing mathematical expressions, LaTeX is used to format the expressions properly. For example, $\sqrt{3}$ is written in LaTeX as $\sqrt{3}$.
In this problem, the process involves multiplying the square root of 3 by the nested radical expression and then representing the result in both exact and decimal forms.
