Simplify square root of 2x( square root of 8x- square root of 32)

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:

1. 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 $\sqrt{a}$.

2. Factoring: Factoring involves writing a number or expression as a product of its factors. For example, $32$ can be factored into $2^5$ or $4^2\cdot 2$.

3. 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.

4. 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.

5. 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 $16$.

6. 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.