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Assuming
x and
y are both positive, write the following expression in simplest radical form.

4x^(2)y^(3)sqrt(27x^(2)y^(7))
Answer:

Assuming $x$ and $y$ are both positive, write the following expression in simplest radical form. $\newline$ $4 x^{2} y^{3} \sqrt{27 x^{2} y^{7}}$ $\newline$ Answer:

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Simplify radical expressions with root inside the root

Full solution

Q. Assuming

x

and

y

are both positive, write the following expression in simplest radical form.

\newline

4 x^{2} y^{3} \sqrt{27 x^{2} y^{7}}

\newline

Answer:

Write Expression: Write the given expression and identify the parts that can be simplified. $\newline$ The given expression is $4x^{2}y^{3}\sqrt{27x^{2}y^{7}}$ . $\newline$ We can simplify the square root part by factoring out perfect squares.
Factor Inside Square Root: Factor the expression inside the square root to identify perfect squares. $\newline$ The expression inside the square root is $27x^{2}y^{7}$ . $\newline$ $27$ is a perfect cube ( $3^{3}$ ), $x^{2}$ is already a perfect square, and $y^{7}$ can be written as $(y^{6}\cdot y)$ , where $y^{6}$ is a perfect square.
Take Out Perfect Squares: Take out the perfect squares from under the square root. $\newline$ $\sqrt{27x^{2}y^{7}} = \sqrt{3^{3} \times x^{2} \times y^{6} \times y}$ $\newline$ = $\sqrt{3^{2} \times 3 \times x^{2} \times y^{6}} \times \sqrt{y}$ $\newline$ = $3xy^{3} \times \sqrt{3y}$
Multiply Simplified Square Root: Multiply the simplified square root by the rest of the given expression. $\newline$ Now we have $4x^{2}y^{3} \times 3xy^3 \times \sqrt{3y}$ . $\newline$ Multiplying the coefficients (numbers in front of the variables) gives us $4 \times 3 = 12$ . $\newline$ Multiplying the variables with the same base, we add the exponents: $x^{2+1} = x^3$ and $y^{3+3} = y^6$ .
Write Final Expression: Write the final simplified expression. $\newline$ The final expression is $12x^3y^6 \cdot \sqrt{3y}$ .

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