Simplify. $\newline$ Remove all perfect squares from inside the square root. Assume $x$ is positive. $\newline$ $\sqrt{54 x^{7}}= \square$

Full solution

Q. Simplify.

\newline

Remove all perfect squares from inside the square root. Assume

x

is positive.

\newline

\sqrt{54 x^{7}}= \square

Factor Perfect Squares: Factor the expression inside the square root to identify perfect squares. $\newline$ We need to factor $54x^7$ into its prime factors and identify perfect squares. $\newline$ $54$ can be factored into $2 \times 3 \times 3 \times 3$ , and $x^7$ is already in prime form as $x \times x \times x \times x \times x \times x \times x$ . $\newline$ So, $\sqrt{54x^7} = \sqrt{2 \times 3^3 \times x^6 \times x}$ .
Group Factors: Group the factors into perfect squares and separate the non-perfect squares. $\newline$ We have $3^3$ which is $3^2 \times 3$ , and $x^6$ which is $(x^3)^2$ . These are our perfect squares. $\newline$ So, $\sqrt{54x^7} = \sqrt{2 \times 3^2 \times x^6 \times 3 \times x}$ .
Take Square Roots: Take the square root of the perfect squares and leave the non-perfect squares inside the square root. $\newline$ The square root of $3^2$ is $3$ , and the square root of $x^6$ is $x^3$ . $\newline$ So, $\sqrt{54x^7} = 3 \times x^3 \times \sqrt{2 \times 3 \times x}$ .
Simplify Expression: Simplify the expression inside the square root. We can't simplify $\sqrt{2 \times 3 \times x}$ any further because there are no more perfect squares. So, the final simplified expression is $3x^3 \times \sqrt{6x}$ .