Answer:
Product of cube root of :
[tex]\sqrt[3]{16x^7} \times \sqrt[3]{12x^9}[/tex]
First simplify :[tex]\sqrt[3]{16x^7}[/tex]
Factor: One of two or more expressions that are multiplied together to get a product
then, we can write it as:
[tex]\sqrt[3]{2 \cdot 8 x^7}[/tex] ,
Rewrite 8 as [tex]2^3[/tex] and [tex]x^7 = x^6 \cdot x[/tex]
[tex]\sqrt[3]{2\cdot 2^3 \cdot x^6 \cdot x}[/tex] or
[tex]\sqrt[3]{2\cdot 2^3 \cdot (x^2)^3 \cdot x}[/tex] [∵ [tex](a^x)^y=a^{xy}[/tex] ]
or [tex]\sqrt[3]{2^3 \cdot (x^2)^3 \cdot 2\cdot x}[/tex]
or [tex]2 \cdot x^2\sqrt[3]{2\cdot x}[/tex] [∵[tex]\sqrt[3]{a^3} =a[/tex] ]
Similarly, we simplify for [tex]\sqrt[3]{12 x^9}[/tex]
Then, we can write it as [tex]\sqrt[3]{12\cdot (x^3)^3}[/tex] or
[tex]x^3 \cdot \sqrt[3]{12}[/tex]
Use : [tex]x^{a+b}=x^a \cdot x^b[/tex] , [tex]\sqrt[3]{a} \cdot\sqrt[3]{b} = \sqrt[3]{a \cdot b}[/tex]
Now,
[tex]\sqrt[3]{16x^7} \times \sqrt[3]{12x^9}[/tex] = [tex]2 \cdot x^2\sqrt[3]{2\cdot x} \times x^3 \cdot \sqrt[3]{12}[/tex]
= [tex]2x^2 \cdot x^3 \sqrt[3]{2x} \cdot \sqrt[3]{12}[/tex]
= [tex]2 x^5\cdot \sqrt[3]{2x \cdot 12}[/tex] or [tex]2x^5 \cdot \sqrt[3]{24 x}[/tex]
= [tex]2x^5 \cdot \sqrt[3]{8 \cdot 3x}[/tex] or [tex]2 x^5 \cdot \sqrt[3]{2^3 \cdot 3 \cdot x}[/tex]
= [tex]2x^5 \cdot 2 \sqrt[3]{3x}[/tex] = [tex]4 x^5 \cdot \sqrt[3]{3x}[/tex]
therefore, the product of [tex]\sqrt[3]{16x^7} \times \sqrt[3]{12x^9}[/tex] is, [tex]4 x^5 \cdot \sqrt[3]{3x}[/tex]
