Full solution

Q. Write the expression in simplest form.

\newline

2 \sqrt{7 x} \cdot 3 \sqrt{14 x^{2}}

Write Expression: Write down the given expression. $\newline$ We are given the expression $2\sqrt{7x} \times 3\sqrt{14x^{2}}$ . We need to multiply these two square roots together.
Apply Property: Apply the multiplication property of square roots. $\newline$ According to the multiplication property of square roots, $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$ . So, we can multiply the square roots together to get $\sqrt{7x} \times \sqrt{14x^{2}}$ .
Multiply Coefficients and Roots: Multiply the coefficients and the square roots separately. $\newline$ The coefficients are $2$ and $3$ , and the square roots are $\sqrt{7x}$ and $\sqrt{14x^{2}}$ . Multiplying the coefficients gives us $2 \times 3 = 6$ . Multiplying the square roots gives us $\sqrt{7x \times 14x^{2}}$ .
Simplify Inside Square Root: Simplify the expression inside the square root. $\newline$ We have $\sqrt{7x \times 14x^{2}}$ which simplifies to $\sqrt{98x^{3}}$ . This is because $7x \times 14x^{2} = 98x^{3}$ .
Factor Out Perfect Squares: Factor out perfect squares from the square root. We notice that $98$ is $49 \times 2$ and $49$ is a perfect square ( $7^2$ ). Also, $x^{3}$ can be written as $x^{2} \times x$ , where $x^{2}$ is a perfect square. So, we can rewrite $\sqrt{98x^{3}}$ as $\sqrt{49 \times 2 \times x^{2} \times x}$ .
Take Square Root of Perfect Squares: Take the square root of the perfect squares. $\newline$ The square root of $49$ is $7$ , and the square root of $x^{2}$ is $x$ . So, we can take these out of the square root to get $7x \sqrt{2 \times x}$ .
Multiply by Coefficient: Multiply the result by the coefficient from Step $3$ . $\newline$ We have $6 \times (7x \times \sqrt{2 \times x})$ , which simplifies to $42x \times \sqrt{2 \times x}$ .