Answer:
[tex]14 ( 5)^{\frac{1}{4}} p^{\frac{1}{4}} q^{\frac{1}{2}} r^{2}[/tex]
Step-by-step explanation:
The expression to simplify in this problem is
[tex]7\sqrt[4]{80pq^2r^8}[/tex]
Which can be rewritten as
[tex]7(80pq^2r^8)^{\frac{1}{4}}[/tex]
Or also as
[tex]7\cdot 80^{\frac{1}{4}} \cdot p^{\frac{1}{4}} \cdot (q^2)^{\frac{1}{4}} \cdot (r^8)^{\frac{1}{4}}[/tex]
Now we can apply the following rule for the calculation of the power of a power:
[tex](a^m)^n = a^{m\cdot n}[/tex]
So we get:
[tex]7\cdot 80^{\frac{1}{4}} \cdot p^{\frac{1}{4}} \cdot q^{\frac{1}{2}} \cdot r^{2}[/tex]
Which can therefore be rewritten as
[tex]7\cdot (2^4\cdot 5)^{\frac{1}{4}} \cdot p^{\frac{1}{4}} \cdot q^{\frac{1}{2}} \cdot r^{2}[/tex]
And so, we get
[tex]7\cdot 2 \cdot ( 5)^{\frac{1}{4}} \cdot p^{\frac{1}{4}} \cdot q^{\frac{1}{2}} \cdot r^{2}[/tex]
which can be finally rewritten as
[tex]14 ( 5)^{\frac{1}{4}} p^{\frac{1}{4}} q^{\frac{1}{2}} r^{2}[/tex]
