Roots and radicals: complete GMAT guide

The square root of a nonnegative number $a$ is the nonnegative number whose square is $a$. The symbol $\sqrt{a}$ denotes the principal, nonnegative root. For example, $\sqrt{25}=5$, not $-5$.

The equation $x^2=25$ has two solutions, $x=5$ and $x=-5$. The square root symbol by itself gives only the nonnegative value. This distinction matters in equations and in Data Sufficiency questions.

Factor out perfect squares from under the radical:

For example, $\sqrt{72}=\sqrt{36 \times 2}=6\sqrt{2}$. Look for the largest perfect square factor. The factors 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 cover most GMAT cases.

To simplify a radical in a fraction, simplify the numerator and denominator separately, then reduce. $\sqrt{18/8}=\sqrt{9/4}=3/2$.

For nonnegative $a$ and $b$:

• $\sqrt{a}\sqrt{b}=\sqrt{ab}$
• $\sqrt{a}/\sqrt{b}=\sqrt{a/b}$, for $b \gt 0$
• $(\sqrt{a})^2=a$

You cannot combine radicals with different radicands by adding them. $\sqrt{2}+\sqrt{3}$ cannot be simplified to $\sqrt{5}$. You can combine like radicals: $3\sqrt{2}+5\sqrt{2}=8\sqrt{2}$.

A simplified answer usually has no radical in the denominator. To remove a single square root, multiply numerator and denominator by that root:

For a denominator of the form $a+\sqrt{b}$, multiply by the conjugate $a-\sqrt{b}$. The product uses the difference of squares:

The conjugate method also works for denominators such as $\sqrt{5}-\sqrt{2}$. Multiply by $\sqrt{5}+\sqrt{2}$.

To solve an equation with a square root, isolate the radical and square both sides. Squaring can introduce extraneous solutions, so check every answer in the original equation.

Always check. Squaring both sides can turn a false statement into a true one, and the GMAT uses this to create trap answers.

The cube root of $a$ is the number whose cube is $a$. Unlike square roots, cube roots of negative numbers are real: $\sqrt[3]{-8}=-2$. The fourth root of a negative number is not real.

The rules for radicals extend to higher roots: $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$ for real roots. To simplify $\sqrt[3]{54}$, factor out $27$: $\sqrt[3]{27 \times 2}=3\sqrt[3]{2}$.

GMAT questions often ask which integer is closest to a root, or whether a root is greater than a certain value. Use perfect squares to bracket the root. Since $7^2=49$ and $8^2=64$, $\sqrt{55}$ is between 7 and 8, closer to 7 because 55 is closer to 49 than to 64.

A square root is the same as a one-half power: $\sqrt{a}=a^{1/2}$. A cube root is $a^{1/3}$. The exponent rules apply:

This connection helps with expressions that mix roots and powers. For example, $\sqrt{x^6}=x^3$ when $x \ge 0$. If $x$ can be negative, the result is $|x|^3$, because the square root is nonnegative.

Square roots appear in geometry questions about right triangles and circles. The Pythagorean theorem is $a^2+b^2=c^2$, so the hypotenuse is $c=\sqrt{a^2+b^2}$. For a right triangle with legs 6 and 8, the hypotenuse is $\sqrt{36+64}=\sqrt{100}=10$. For legs 1 and 1, the hypotenuse is $\sqrt{2}$, which cannot be simplified further.

Some right triangles appear often enough that their side ratios are worth recognizing. A 3-4-5 triangle has sides in the ratio 3:4:5. A 5-12-13 triangle has sides in the ratio 5:12:13. A 45-45-90 triangle has sides in the ratio $1:1:\sqrt{2}$. A 30-60-90 triangle has sides in the ratio $1:\sqrt{3}:2$. Recognizing these ratios can save several steps.

Roots also appear in formulas for distance and standard deviation. The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is

This is the Pythagorean theorem applied to the horizontal and vertical differences. If the differences are 3 and 4, the distance is 5. If they are 1 and 1, the distance is $\sqrt{2}$.

In statistics, standard deviation involves the square root of the average squared deviation from the mean. GMAT questions rarely require the full calculation, but they may ask how the standard deviation changes when every value is multiplied by a constant. Multiplying every value by $k$ multiplies the standard deviation by $|k|$, because the deviations are scaled by $k$ and the square root preserves the absolute value.

Some questions ask for an exact radical, and others ask which choice is closest. Read the answer choices to decide. If the choices are integers or simple fractions, estimate. If the choices contain radicals, simplify exactly.

To estimate a square root, bracket it between perfect squares. $\sqrt{30}$ is between 5 and 6 because $25 \lt 30 \lt 36$. To refine, compare 30 with the midpoint 30.25, which is $5.5^2$. Since 30 is slightly less than 30.25, $\sqrt{30}$ is slightly less than 5.5. This level of precision is usually enough to eliminate choices.

1. Check whether the question asks for the principal root or all solutions.
2. Factor out perfect squares or cubes to simplify.
3. Combine like radicals only.
4. Rationalize the denominator if the answer requires it.
5. For radical equations, square both sides and check for extraneous solutions.
6. Use perfect squares to estimate when an exact value is not needed.

• Writing $\sqrt{25}=\pm 5$. The symbol gives the nonnegative root.
• Adding radicals with different radicands.
• Forgetting to check for extraneous solutions.
• Assuming $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$.
• Ignoring the sign restriction on the square root of a variable expression.
• Rationalizing incorrectly by multiplying only the numerator.
• Confusing the cube root of a negative number with an undefined value.