Square Roots and Radicals

The square root of a number $a$ , written $\sqrt{a}$ , is the non-negative number that, when multiplied by itself, gives $a$ . For example, $\sqrt{25} = 5$ because $5 \times 5 = 25$ . The symbol $\sqrt{\phantom{x}}$ is called a radical, and the number inside it is the radicand.

Perfect Squares

A perfect square is a number that is the square of an integer. Memorizing the first several perfect squares makes radical problems much faster:

$1^2=1$ , $2^2=4$ , $3^2=9$ , $4^2=16$ , $5^2=25$
$6^2=36$ , $7^2=49$ , $8^2=64$ , $9^2=81$ , $10^2=100$
$11^2=121$ , $12^2=144$ , $13^2=169$ , $14^2=196$ , $15^2=225$

Recognizing these instantly (in both directions — knowing $144$ is a perfect square and that its root is $12$ ) saves significant time on contests.

Simplifying Radicals

When the radicand isn't a perfect square, we simplify by factoring out the largest perfect square that divides it, using the rule $\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}$ . For example, to simplify $\sqrt{72}$ : notice $72 = 36 \times 2$ , where $36$ is the largest perfect square factor. Then: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36}\cdot\sqrt{2} = 6\sqrt{2}$ Always double-check you pulled out the largest perfect square factor — using a smaller one (like $4$ instead of $36$ ) still works but takes an extra simplification step.

Estimating Non-Perfect-Square Roots

For a number that isn't a perfect square, find the two perfect squares it falls between to estimate the root. To estimate $\sqrt{50}$ : since $49 = 7^2$ and $64=8^2$ , and $50$ is just barely more than $49$ , $\sqrt{50}$ is just a bit more than $7$ — in fact, $\sqrt{50} \approx 7.07$ . This estimation skill is useful whenever a problem asks which integer a radical is closest to, or for a quick sanity check on a computed answer.

Operations With Radicals

Radicals with the same radicand combine like like terms: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$ , but $3\sqrt{2}+5\sqrt{3}$ cannot be combined further since the radicands differ. Multiplication of radicals follows $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$ , so $\sqrt{3}\cdot\sqrt{12}=\sqrt{36}=6$ .

Example

Simplify $\sqrt{98}$ and then find $\sqrt{98} + \sqrt{2}$ .

Step 1: Factor out the largest perfect square from $98$ : $98 = 49\times2$ , and $49$ is a perfect square.

Step 2: $\sqrt{98} = \sqrt{49}\cdot\sqrt{2} = 7\sqrt{2}$ .

Step 3: Now add: $7\sqrt{2} + \sqrt{2} = 8\sqrt{2}$ , since both terms share the radicand $2$ .

Key Takeaways

$\sqrt{a}$ is the non-negative number whose square is $a$ ; memorize perfect squares through at least $15^2$ .
Simplify radicals by factoring out the largest perfect square: $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ .
Estimate non-perfect-square roots by finding the nearest perfect squares above and below.
Only radicals with matching radicands can be combined by addition/subtraction.

Practice problems

2002 AMC 8 Problem 6 AMC 8

1 pts

2002 AMC 8 Problem 7 AMC 8

1 pts

2002 AMC 8 Problem 8 AMC 8

1 pts

2002 AMC 8 Problem 9 AMC 8

1 pts

2002 AMC 8 Problem 10 AMC 8

1 pts