Simplifying and Combining Surds

Simplifying and Combining Surds

A surd is an exact root left in radical form. Simplify by taking perfect-square factors out of a square root, then combine only like surds. For example, $\sqrt{72}=\sqrt{36\times2}=6\sqrt2$. Like $3\sqrt2$ and $5\sqrt2$ can be added because they have the same irrational part; $\sqrt2$ and $\sqrt3$ cannot.

Worked reasoning

$2\sqrt{18}-\sqrt8=2(3\sqrt2)-2\sqrt2=4\sqrt2$. First simplify both radicals; only then collect like terms.

Exam method

1. Factor each radicand into a largest square factor times a remainder. 2. Simplify every radical. 3. Combine coefficients of matching surds only.

Exact-answer discipline

A decimal approximation and a simplified surd answer different questions. Leave a radical exact unless the question says to round. To decide whether a root can simplify, search for the largest square factor: 108 has 36, so $\sqrt{108}=6\sqrt3$. At the end, scan all terms once more. If two radical parts match, combine their coefficients; if they do not match, the expression is already as collected as it can be.

One-minute retrieval: Simplifying and Combining Surds

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.