Algebra

4th_root(7a^3) * 4th_root(343a^2)

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Rewrite using fourth root notation

We start by recognizing both terms are fourth roots. We can use the product rule for radicals, which says two roots with the same index can be multiplied together under one radical.
2

Combine under one fourth root

The product rule states . So we multiply the expressions inside together under a single fourth root.
3

Multiply the values inside the radical

and by the rule of adding exponents. Note that .
4

Rewrite 2401 as a power of 7

Since , we rewrite it this way so we can easily pull it out of the fourth root.
5

Split a^5 into a^4 times a

We break into so that is a perfect fourth power and can come out of the radical.
6

Pull perfect fourth powers outside the radical

and , so both come out as whole factors. The leftover stays inside the fourth root.

Understanding this problem

Learning Insight

The product rule for radicals works because radicals are just fractional exponents: . Multiplying two fourth roots is the same as multiplying the bases and keeping the exponent . Recognizing perfect powers (like ) is the key skill that makes simplification possible.

Quick Tip

When you see and together, notice , so — a perfect fourth power that pops right out of a fourth root!

Common Mistake

Students often add the numbers inside instead of multiplying: writing instead of . Always MULTIPLY when combining things under one radical using the product rule.