Select the correct answer. If $$ x0 $$ and $$ y0 $$, which expression is equivalent to $$ \sqrt{768 x^{19} y^{37}} ? $$ A. $$ 8 x^{9} y^{18} \sqrt{12 x y} $$ B. $$ 16 x^{9} y^{18} \sqrt{3 x y} $$ C. $$ 16 x^{4} y^{6} \sqrt{3 x^{4} y} $$ D. $$ 8 x^{4} y^{6} \sqrt{12 x^{4} y} $$

Question

Select the correct answer. If $$ x>0 $$ and $$ y>0 $$, which expression is equivalent to $$ \sqrt{768 x^{19} y^{37}} ? $$ A. $$ 8 x^{9} y^{18} \sqrt{12 x y} $$ B. $$ 16 x^{9} y^{18} \sqrt{3 x y} $$ C. $$ 16 x^{4} y^{6} \sqrt{3 x^{4} y} $$ D. $$ 8 x^{4} y^{6} \sqrt{12 x^{4} y} $$

UpStudy AI · Accepted Answer

We start with the expression

$$
\sqrt{768 x^{19} y^{37}}.
$$

**Step 1. Factor the constant.**

Notice that

$$
768=256\cdot 3,
$$

and since

$$
256=2^8,
$$

we have

$$
768=2^8\cdot 3.
$$

**Step 2. Express the exponents in the radicand as perfect squares times a leftover.**

For the variable $$x^{19}$$, write the exponent as

$$
19=18+1,
$$

so

$$
x^{19}=x^{18}\cdot x.
$$

For the variable $$y^{37}$$, write the exponent as

$$
37=36+1,
$$

so

$$
y^{37}=y^{36}\cdot y.
$$

**Step 3. Rewrite the entire expression under the square root.**

$$
\sqrt{768 x^{19} y^{37}} = \sqrt{2^8\cdot 3\cdot x^{18}\cdot x\cdot y^{36}\cdot y}.
$$

**Step 4. Separate the square root into parts where perfect squares can be taken out.**

$$
\sqrt{2^8\cdot x^{18}\cdot y^{36}} = 2^4 x^9 y^{18} = 16 x^9 y^{18},
$$

and the remaining factors stay under the square root:

$$
\sqrt{3xy}.
$$

Thus, the expression becomes

$$
16 x^9 y^{18}\sqrt{3xy}.
$$

**Step 5. Identify the correct answer.**

The correct answer is

$$
\textbf{B. } 16x^9y^{18}\sqrt{3xy}.
$$
The correct answer is $$ 16x^{9}y^{18}\sqrt{3xy} $$, choice B.

Select the correct answer.
If and , which expression is equivalent to
A.
B.
C.
D.

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