Select the correct answer. Which expression is equivalent to $$ 7 x^{2} \sqrt{2 x^{4}} \cdot 6 \sqrt{2 x^{12}} $$, if $$ x
eq 0 $$ ? A. $$ 26 x^{22} $$ B. $$ 13 x^{12} \sqrt{2} $$ C. $$ 84 x^{10} $$ D. $$ 42 x^{12} \sqrt{2} $$

Question

Select the correct answer. Which expression is equivalent to $$ 7 x^{2} \sqrt{2 x^{4}} \cdot 6 \sqrt{2 x^{12}} $$, if $$ x 
eq 0 $$ ? A. $$ 26 x^{22} $$ B. $$ 13 x^{12} \sqrt{2} $$ C. $$ 84 x^{10} $$ D. $$ 42 x^{12} \sqrt{2} $$

UpStudy AI · Accepted Answer

We start with the given expression:
$$
7x^{2} \sqrt{2x^{4}} \cdot 6 \sqrt{2x^{12}}
$$

**Step 1.** Rewrite the square roots by separating the factors under the square root:
$$
\sqrt{2x^4} = \sqrt{2} \cdot \sqrt{x^4} \quad \text{and} \quad \sqrt{2x^{12}} = \sqrt{2} \cdot \sqrt{x^{12}}
$$

Since $$ \sqrt{x^4} = x^2 $$ and $$ \sqrt{x^{12}} = x^6 $$ (because $$ x \neq 0 $$), the expressions simplify to:
$$
\sqrt{2x^4} = \sqrt{2} \, x^2 \quad \text{and} \quad \sqrt{2x^{12}} = \sqrt{2} \, x^6
$$

**Step 2.** Substitute the simplified forms back into the original expression:
$$
7x^{2} \cdot \left(\sqrt{2} \, x^2\right) \cdot 6 \cdot \left(\sqrt{2} \, x^6\right)
$$

**Step 3.** Combine like terms:
- Multiply the numerical constants:
$$
7 \times 6 = 42
$$
- Multiply the square roots of 2:
$$
\sqrt{2} \cdot \sqrt{2} = 2
$$
- Multiply the powers of $$ x $$:
$$
x^2 \cdot x^2 \cdot x^6 = x^{2+2+6} = x^{10}
$$

**Step 4.** Combine all factors:
$$
42 \times 2 \times x^{10} = 84x^{10}
$$

The expression is equivalent to:
$$
84x^{10}
$$

Hence, the correct answer is Option C.
The expression is equivalent to $$ 84x^{10} $$, so the correct answer is C.

Select the correct answer.
Which expression is equivalent to , if ?
A.
B.
C.
D.

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