Simplify square root of 3 square root of 4 square root of x
The question asks to perform a simplification on a mathematical expression involving square roots. It involves three square roots multiplied together: the square root of 3, the square root of 4, and the square root of x. The task is to combine or simplify these square roots if possible, applying the properties of square roots and multiplication to express the result in a more simplified or elementary form. The goal is not to find the numerical value of x but rather to reduce the expression into simpler terms, assuming that x represents any non-negative real number.
$\sqrt{3 \sqrt{4 \sqrt{x}}}$
Answer
Solution:
Step 1:
Express the number 4 as a power of 2: $2^2$. Thus, the expression becomes $\sqrt{3 \sqrt{2^{2} \sqrt{x}}}$.
Step 2:
Extract the square root of the perfect square from under the radical sign: $\sqrt{3 \cdot 2 \sqrt{\sqrt{x}}}$.
Step 3:
Convert the nested square root of x into a fourth root: $\sqrt{3 \cdot 2 \sqrt[4]{x}}$.
Step 4:
Combine the constants outside the radical: $\sqrt{6 \sqrt[4]{x}}$.
Knowledge Notes:
To simplify an expression involving multiple nested radicals, we can follow these steps:
Identify and rewrite any perfect squares under the radical as powers of their square roots. For example, $4$ can be written as $2^2$.
For any radical within a radical (nested radical), if the innermost expression is a perfect square, you can pull the square root of that number out from under the radical. This simplifies the expression.
When dealing with nested square roots, such as $\sqrt{\sqrt{x}}$, we can rewrite this as a single radical with an index that is the product of the indices of the nested radicals. In this case, two square roots give us a fourth root, so $\sqrt{\sqrt{x}}$ becomes $\sqrt[4]{x}$.
Finally, simplify the expression by multiplying any constants outside the radical together.
Relevant mathematical properties used in this process include:
The property of radicals that states $\sqrt[n]{a^n} = a$, where $n$ is a positive integer and $a$ is a non-negative real number.
The property of radicals that allows us to combine radicals with the same index: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.
The understanding of nested radicals and how to simplify them by increasing the index of the radical.
