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Which expression is equivalent to [tex]$7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}$[/tex], if [tex]$x \neq 0$[/tex]?

A. [tex][tex]$26 x^{22}$[/tex][/tex]

B. [tex]$13 x^{12} \sqrt{2}$[/tex]

C. [tex]$84 x^{10}$[/tex]

D. [tex]$42 x^{12} \sqrt{2}$[/tex]

Sagot :

GinnyAnswer
To simplify the expression [tex]\(7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}\)[/tex], we need to break down each part and simplify step-by-step.

1. Simplify [tex]\( \sqrt{2 x^4} \)[/tex]:
[tex]\[ \sqrt{2 x^4} = \sqrt{2} \cdot \sqrt{x^4} = \sqrt{2} \cdot x^2 \][/tex]

2. Substitute this back into the expression:
[tex]\[ 7 x^2 \sqrt{2 x^4} = 7 x^2 \cdot (\sqrt{2} \cdot x^2) = 7 \sqrt{2} \cdot x^4 \][/tex]

3. Simplify [tex]\( \sqrt{2 x^{12}} \)[/tex]:
[tex]\[ \sqrt{2 x^{12}} = \sqrt{2} \cdot \sqrt{x^{12}} = \sqrt{2} \cdot x^6 \][/tex]

4. Substitute this back into the expression:
[tex]\[ 6 \sqrt{2 x^{12}} = 6 \cdot (\sqrt{2} \cdot x^6) = 6 \sqrt{2} \cdot x^6 \][/tex]

5. Multiply the simplified terms together:
[tex]\[ (7 \sqrt{2} \cdot x^4) \cdot (6 \sqrt{2} \cdot x^6) \][/tex]

6. Combine the coefficients and the square roots:
[tex]\[ 7 \cdot 6 \cdot (\sqrt{2} \cdot \sqrt{2}) \cdot x^{4+6} \][/tex]
[tex]\[ = 42 \cdot 2 \cdot x^{10} \][/tex]
[tex]\[ = 84 x^{10} \][/tex]

Therefore, the expression [tex]\(7 x^2 \sqrt{2 x^4} \cdot 6 \sqrt{2 x^{12}}\)[/tex] simplifies to [tex]\(84 x^{10}\)[/tex].

The correct answer is:
C. [tex]\(84 x^{10}\)[/tex]
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