Respuesta :

gmany
Use:
[tex]\sqrt[n]{a}\cdot\sqrt[n]{b}=\sqrt[n]{a\cdot b}\\\\a^n\cdot a^m=a^{n+m}\\\\(a^n)^m=a^{n\cdot m}\\\\\sqrt[n]{a^n}=a[/tex]


[tex]\sqrt[3]{2x^5}\cdot\sqrt[3]{64x^9}=\sqrt[3]{2x^5}\cdot\sqrt[3]{64}\cdot\sqrt[3]{x^9}\\\\=\sqrt{2x^5}\cdot4\cdot\sqrt[3]{x^9}=4\sqrt[3]{2x^5\cdot x^9}\\\\=4\sqrt[3]{2x^{5+9}}=4\sqrt[3]{2x^{14}}=4\sqrt[3]{2x^{2+12}}=4\sqrt[3]{2x^2x^{12}}\\\\=4\sqrt[3]{2x^2x^{4\cdot3}}=4\sqrt[3]{2x^2(x^4)^3}=4\sqrt[3]{2x^2}\cdot\sqrt[3]{(x^4)^3}\\\\=4\sqrt[3]{2x^2}\cdot x^4=\boxed{4x^4\sqrt[3]{2x^2}}[/tex]
facundo3141592

We want to get the simplified product for the given expression. The correct option is:

[tex] 4*x^4*\sqrt[3]{2x^{2}}[/tex]

Working with exponents.

First, remember two rules:

[tex]x^a*x^b = x^{a + b}\\ \\ \sqrt{x} *\sqrt{y} = \sqrt{x*y} [/tex]

Now we start with our product:

[tex]\sqrt[3]{2x^5} *\sqrt[3]{64x^9}[/tex]

Using the second rule we get:

[tex]\sqrt[3]{2x^5*64x^9}[/tex]

And using the first one:

[tex]\sqrt[3]{128x^{14}}[/tex]

Now we need to simplify this.

Now we will use the rules in inverse order, we can rewrite:

[tex]\sqrt[3]{128x^{14}} = \sqrt[3]{128x^{12}*x^2} = \sqrt[3]{128x^{2}}*\sqrt[3]{x^{12}} \\ \\ \sqrt[3]{128x^{2}}*\sqrt[3]{x^{12}} = \sqrt[3]{128x^{2}}*x^{12/3} = \sqrt[3]{128x^{2}}*x^4[/tex]

We already simplified the variable part, now we need to simplify the number.

128 = 2*64

And:

4*4*4 = 64

Then:

[tex] \sqrt[3]{128x^{2}}*x^4 = \sqrt[3]{2x^{2}}*\sqrt[3]{64} *x^4 = 4*\sqrt[3]{2x^{2}}*x^4[/tex]

From this, we can conclude that the correct option is the third one.

If you want to learn more about exponents, you can read:

https://brainly.com/question/11464095

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