Roots

If you’re comfortable with exponents, you’ll find roots easy to understand. Roots are essentially the reverse of exponents. The square root of a number is a value that, when squared, gives that number. For example, both 3 and -3 squared give 9, so the square roots of 9 are 3 and -3. Simple, right?

Just like there are exponents of various powers, there are corresponding roots. While the GMAT often focuses on square roots, you might also encounter cube roots, fourth roots, and others. These roots are identified by the radical sign (√). For even roots, such as square roots, only the positive root is typically given. For instance, the square root of 9 is 3.

Consider the cube root of 27, expressed as $\sqrt[3]{27}$. This asks what number, when raised to the third power, equals 27. Since $3^3 = 27$, the cube root of 27 is 3.

Radicals, even when they look complicated, can often be simplified. For example, if you encounter $\sqrt{98}$, it’s not the final answer. Look for perfect square factors of 98. Since 98 can be factored as $98 = 49 \times 2$, and 49 is a perfect square ($\sqrt{49} = 7$), you can simplify it to $7 \sqrt{2}$.

Here's a GMAT-style question that involves roots:

If $n \sqrt[4]{512} = \sqrt[4]{2} n$, then what is the value of $n$?

(A) 1

(B) 1/2

(C) 1

(D) 4

(E) 8

To solve this, simplify the radical. The fourth root of 512 is equal to 4 times the fourth root of 2. Factor 512 as $512 = 256 \times 2$, so the fourth root of 512 becomes $\sqrt[4]{256 \times 2} = \sqrt[4]{256} \times \sqrt[4]{2} = 4 \sqrt[4]{2}$. Since $\sqrt[4]{256} = 4$, it follows that $n = 4$, and the correct answer is (D).

Roots follow the same rules as exponents when performing operations. For example, you can add or subtract roots if they are of the same order and contain the same number. Here are some examples:

• $5 \sqrt{7} + 6 \sqrt{7} = 11 \sqrt{7}$
• $11 \sqrt{a} - 6 \sqrt{a} = 5 \sqrt{a}$

When multiplying or dividing roots, ensure the roots are of the same order. For multiplication, multiply the values inside the radicals:

• $\sqrt{9} \times \sqrt{27} = \sqrt{9 \times 27} = \sqrt{243}$

For division, divide the values inside the radicals:

• $\sqrt{9} \div \sqrt{3} = \sqrt{9 \div 3} = \sqrt{3}$

Here’s a GMAT-style question on operations with radicals:

$\sqrt{16 + 9}$ ?

(A) 5

(B) 7

(C) 12

(D) 25

(E) 625

Be careful with how the radical is written. The square root symbol covers the entire expression, so add the numbers under the radical first. $16 + 9 = 25$, and $\sqrt{25} = 5$. So, the correct answer is (A). If you got 7, it means you mistakenly took the square root of each term separately before adding them.