Simplify square root of 2x^2* square root of 15x^2

Solution:

Step 1:

Arrange the factors within the square roots. $\sqrt{2x^2}\cdot \sqrt{15x^2}$

Step 2:

Extract the square root of $x^2$ from the first radical. $x\sqrt{2}\cdot \sqrt{15x^2}$

Step 3:

Arrange the factors within the second square root. $x\sqrt{2}\cdot \sqrt{x^2\cdot 15}$

Step 4:

Extract the square root of $x^2$ from the second radical. $x\sqrt{2}\cdot (x\sqrt{15})$

Step 5:

Combine the $x$ terms by using the properties of exponents.

Step 5.1:

Rearrange the terms. $x\cdot x\sqrt{2}\cdot \sqrt{15}$

Step 5.2:

Combine the $x$ terms. $x^2\sqrt{2}\cdot \sqrt{15}$

Step 6:

Multiply the radicals $x^2\sqrt{2}$ and $\sqrt{15}$.

Step 6.1:

Apply the product rule for radicals. $x^2\sqrt{2\cdot 15}$

Step 6.2:

Calculate the product inside the radical. $x^2\sqrt{30}$

Knowledge Notes:

To simplify the expression $\sqrt{2x^2}\cdot \sqrt{15x^2}$, we use several properties of square roots and exponents:

1. Product Property of Square Roots: $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$, which allows us to combine or separate square roots.

2. Square Root of a Square: $\sqrt{x^2}=x$, assuming $x$ is non-negative. This is because squaring a number and then taking the square root should return the original non-negative number.

3. Multiplication of Radicals: When multiplying radicals, we can multiply the numbers inside the radicals together under a single radical sign.

4. Exponent Rules: When multiplying like bases, we add the exponents: $x^m\cdot x^n=x^{m+n}$.

Using these properties, we can simplify the given expression step by step, extracting square roots of perfect squares and combining radicals until we reach the simplest form.