Simplify square root of 2x( square root of 8x- square root of 32)
The problem asks to simplify a given mathematical expression that involves square roots and variables. The expression contains nested square roots and requires applying the properties of radicals and combining like terms. Simplification would typically involve multiplying the square roots, distributing the multiplication over subtraction, and simplifying any square roots of squared terms that can be represented as the original variable or number without the radical sign.
$\sqrt{2 x \left(\right. \sqrt{8 x - \sqrt{32}} \left.\right)}$
Answer
Solution:
Step:1
Express $32$ in terms of its prime factors as $2^5$.
Step:1.1
Extract the square root of $16$ from $32$ as $\sqrt{2x\sqrt{8x - \sqrt{16 \cdot 2}}}$.
Step:1.2
Represent $16$ using its base and exponent as $\sqrt{2x\sqrt{8x - \sqrt{(2^4) \cdot 2}}}$.
Step:2
Simplify the square roots by taking out the square terms $\sqrt{2x\sqrt{8x - 4\sqrt{2}}}$.
Step:3
Combine like terms by multiplying $4$ by $-1$ to get $\sqrt{2x\sqrt{8x - 4\sqrt{2}}}$.
Step:4
Extract the common factor of $4$ from $8x - 4\sqrt{2}$.
Step:4.1
Take $4$ out of $8x$ as $\sqrt{2x\sqrt{4(2x) - 4\sqrt{2}}}$.
Step:4.2
Take $4$ out of $-4\sqrt{2}$ as $\sqrt{2x\sqrt{4(2x) + 4(-\sqrt{2})}}$.
Step:4.3
Factor out $4$ from $4(2x) + 4(-\sqrt{2})$ to get $\sqrt{2x\sqrt{4(2x - \sqrt{2})}}$.
Step:5
Express $4$ as the square of $2$, written as $\sqrt{2x\sqrt{(2^2)(2x - \sqrt{2})}}$.
Step:6
Extract the square term from under the radical to obtain $\sqrt{2x \cdot 2\sqrt{2x - \sqrt{2}}}$.
Step:7
Multiply $2$ by the square root of $x$ to get $\sqrt{4x\sqrt{2x - \sqrt{2}}}$.
Step:8
Rewrite $4x\sqrt{2x - \sqrt{2}}$ as the product of $2^2$ and the remaining terms.
Step:8.1
Represent $4$ as $2^2$ to get $\sqrt{(2^2)x\sqrt{2x - \sqrt{2}}}$.
Step:8.2
Enclose the terms in parentheses to form $\sqrt{(2^2)(x\sqrt{2x - \sqrt{2}})}$.
Step:9
Finally, take the square term out from under the radical to arrive at $2\sqrt{x\sqrt{2x - \sqrt{2}}}$.
Knowledge Notes:
Square roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of $a$ is denoted as $\sqrt{a}$.
Factoring: Factoring involves writing a number or expression as a product of its factors. For example, $32$ can be factored into $2^5$ or $4^2 \cdot 2$.
Simplifying square roots: To simplify a square root, one can extract square factors from under the radical sign. For instance, $\sqrt{16 \cdot 2}$ simplifies to $4\sqrt{2}$ because $16$ is a perfect square.
Combining like terms: This involves adding or subtracting terms that have the same variable raised to the same power. For example, $4x - 4\sqrt{2}$ has a common factor of $4$ that can be factored out.
Exponentiation: An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, $2^4$ means $2$ multiplied by itself $4$ times, which equals $16$.
Radical expressions: A radical expression is an expression that contains a radical symbol (鈭?. Simplifying radical expressions often involves factoring out perfect squares and combining like terms.
