Algebra

2a*sqrt(125a^2*b) - a^2*sqrt(80b) + 4*sqrt(20a^4*b)

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Simplify the first radical term

Break 125 into 25·5 and pull out the perfect square. . So the first term becomes .
2

Simplify the second radical term

Break 80 into 16·5 and pull out the perfect square. . So the second term becomes .
3

Simplify the third radical term

Break 20 into 4·5 and pull out perfect squares. . So the third term becomes .
4

Rewrite the full expression with simplified terms

Replace each original term with its simplified form. All three terms now share the same radical , so they are like terms and can be combined.
5

Combine like terms by adding coefficients

Factor out the common part and add the numbers: . The simplified result is .

Understanding this problem

Learning Insight

Radical expressions work just like variable terms — you can only add or subtract them when the part inside the radical (the radicand) is exactly the same. Simplifying each radical first is the key step that reveals the hidden like terms. This is the same idea as combining into .

Quick Tip

Always simplify radicals completely before trying to combine terms. Look for the largest perfect-square factor inside each radical — that's what you pull out front.

Common Mistake

Students often forget to also bring variables outside the radical. For example, (assuming ), so loses the from under the radical — forgetting that step leaves an incorrect radicand.