Radicals
Simplify radicals
ALEKS placement
Pull out perfect-square factors; what comes out is a root, what stays is not.
What this covers
√ab=√a√b- A square root undoes squaring: √49 is 7 because 7·7 = 49. The number under the bar is called the radicand.
- A perfect square is a number whose square root is a whole number, such as 1, 4, 9, 16, 25, or 36.
- Find the largest perfect square dividing the radicand, split the radical into that square times the rest, and take its root outside.
- Cannot spot a square factor? Split the radicand into primes and circle pairs. Each pair sends one copy outside; leftovers stay under the bar.
- For variables, count copies the same way: x⁵ is five x's, so two pairs leave as x² and one x stays under the bar.
- Check: square the number outside and multiply by what stayed inside. You should get the original radicand back.
Worked example
Worked example
Simplify √72.
- √72=√36⋅2=√36⋅√236 is the largest perfect square that divides 72, since 72 = 36 · 2.
- 6√2√36 = 6 comes outside. 2 has no square factor, so it stays under the bar. Check: 6² · 2 = 72.
Another worked example
Simplify √(72x²) for real x.
- √72x2=√36√2√x2All three factors are nonnegative, so the real product rule is legal.
- 6∣x∣√2The principal square root of x² is |x|, including when x is negative.
A common mistake
A mistake Lemma catches
From √36x4√2x
6x4√2x6x2√2x
“The square root applies to x⁴ as well as to 36: √(x⁴) = x², because x² · x² = x⁴. Only the 36 was rooted here (√36 = 6), and x⁴ was left as it was.”
Try one
Sample problem
Simplify the radical.
√343Practice simplify radicals free
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