Exponents and Roots

Square Roots

Evaluate perfect square roots and simplify non-perfect roots by factoring out perfect squares.

Example

Use perfect-square factors to evaluate and simplify square roots.

highlighted = computed this step

Step 1 — Perfect square

Set up the expression.

64\sqrt{64}
sqrt(64) asks for the side lengthA square root is the side length of a square area.

Step 2 — Check the square

Check the square: 8^2 = 64.

82=64\hlmath{8^{2}} = 64
8 x 8 = 6488The square has 64 tiles and side length 8.

Step 3 — Take the square root

Take the square root: sqrt(64) = 8.

64=8\sqrt{64} = \hl{8}
sqrt(64) = 888The square has 64 tiles and side length 8.

Step 4 — Simplify a radical

Set up the expression.

50\sqrt{50}
Look for a square factor inside sqrt(50)square factor 25leftover 2Split 50 into a square factor and a leftover factor.

Step 5 — Factor the radicand

Factor the radicand: 50 = 25 x 2.

50=25×250 = \hl{25} \times \hl{2}
50 = 25 x 2square factor 25leftover 2The radicand is 25 x 2.

Step 6 — Root the square factor

Take out the square factor: sqrt(25) = 5.

25=5\sqrt{25} = \hl{5}
sqrt(25) = 5; sqrt(2) stayssquare factor 25leftover 2sqrt(25) = 5sqrt(2) staysOnly the square factor comes out of the root.

Step 7 — Result

Keep the leftover under the root: 5sqrt(2).

50=52\sqrt{50} = \hlmath{5\sqrt{2}}
sqrt(50) = 5sqrt(2)square factor 25leftover 2sqrt(25) = 5sqrt(2) staysThe simplified form keeps sqrt(2) symbolic: 5sqrt(2).
square-roots To simplify √n: - If n is a perfect square, state √n = k. - Otherwise factor n = a · b where a is the largest perfect square factor, then √n = √a · √b = k√b.