Answer: [tex]3ab\sqrt[3]{b^4}[/tex]
Step-by-step explanation:
Given the following expression:
[tex]\sqrt[3]{27a^3b^7}[/tex]
You need to apply the Product of powers property, which states that:
[tex](a^m)(a^n)=a^{(m+n)[/tex]
Then, you can rewrite the expression as following:
[tex]=\sqrt[3]{27a^3b^4b^3}[/tex]
The next step is to descompose 27 into its prime factors:
[tex]27=3*3*3=3^3[/tex]
Now you must substitute [tex]3^3[/tex] inside the given root. Then:
[tex]=\sqrt[3]{3^3a^3b^4b^3}[/tex]
You need to remember that, according to Radicals properties:
[tex]\sqrt[n]{a^n}=a^{\frac{n}{n}}=a^1=a[/tex]
Therefore, the final step is to apply this property in order to finally get the expression is its simplest form. This is:
[tex]=3^{\frac{3}{3}}a^{\frac{3}{3}}b^{\frac{4}{3}}b^{\frac{3}{3}}=3ab^{\frac{4}{3}}b=3ab\sqrt[3]{b^4}[/tex]
