# Simplifying Radicals – Techniques & Examples

The word radical in Latin and Greek means “root” and “branch,” respectively. The idea of radicals can be attributed to exponentiation or raising a number to a given power.

The concept of radical is mathematically represented as x n. This expression tells us that a number x is multiplied by itself n number of times. For instance,

3 2 = 3 × 3 = 9, and 2 4 = 2 × 2 × 2 × 2 = 16.

A radical can be defined as a symbol that indicate the root of a number. Square root, cube root, fourth root are all radicals.

The following are the steps required for simplifying radicals:

• Start by finding the prime factors of the number under the radical. Divide the number by prime factors such as 2, 3, 5 until only the left numbers are prime.
• Determine the index of the radical. The index of the radical tells the number of times you need to remove the number from inside to outside radical.
• Move only variables that make groups of 2 or 3 from inside to outside radicals.
• Simplify the expressions both inside and outside the radical by multiplying.
• Simplify by multiplication of all variables both inside and outside the radical.

Example 1

Simplify: √252

Solution

• Find the prime factors of the number inside the radical.

252 = 2 x 2 x 3 x 3 x 7

• Find the radical index, and for this case, our index is two because it is a square root. Therefore, we need two of a kind.

√ (2 x 2 x 3 x 3 x 7)

• Now pull each group of variables from inside to outside the radical. In this case, the pairs of 2 and 3 are moved outside.

2 x 3 √7

• By multiplication, simplify both the expression inside and outside the radical to get the final answer as:

6 √7

Example 2

Simplify:

3√(-432x 7 y 5)

Solution

• To solve such a problem, first, determine the prime factors of the number inside the radical.

432 = 2 x 2 x 2 x2 x 3 x 3 x 3

• Because, it is cube root, then our index is 3.

3√(2 x 2 x 2 x2 x 3 x 3 x 3 x x 7 x y 5)

• Extract each group of variables from inside the radical, and these are 2, 3, x, and y.

-2 x 3 x y 3 x x√(2xy 2)

• Multiply the variables both outside and inside the radical.

-6xy 3√(2xy 2)

Example 3

Find the value of a number n if the square root of the sum of the number with 12 is 5.

Solution

• Write an expression of this problem, square root of the sum of n and 12 is 5
√(n + 12) = square root of the sum.

√(n + 12)=5

• Our equation which should be solved now, is:

√(n + 12) = 5

• On each side the equation is squared:

[√(n + 12)]² = 5²
[√(n + 12)] x [√(n + 12)] = 25
√[(n + 12) x √(n + 12)] = 25
√(n + 12)² = 25
n + 12 = 25

• Subtract 12 from both sides of the expression

n + 12 – 12 = 25 – 12
n + 0 = 25 – 12
n = 13

### Practice Questions

1. Which of the following shows the exponential form of $\sqrt[7]{y}$?

2. Which of the following shows the exponential form of $\sqrt[3]{x^2}$?

3. Which of the following shows the exponential form of $\sqrt[6]{ab}$?

4. Which of the following shows the exponential form of $\sqrt{w^2v^3}$?

5. Which of the following shows the simplified form of $\sqrt[3]{x^8}$?

6. Which of the following shows the simplified form of $\sqrt{8y^3}$?

7. Which of the following shows the simplified form of $\sqrt{x} (4 − 3\sqrt{x})$?

8. Which of the following shows the simplified form of $(\sqrt{x} + 1) (3 − \sqrt[4]{x})$?

9. A rectangular mat is $4$ meters in length and $\sqrt{x + 2}$ meters in width. Which of the following shows the value of $x$ if the perimeter is $24$ meters.

10. Each side of a cube is $5$ meters. A spider connects from the top of the corner of the cube to the opposite bottom corner. What is the total length of the spider web?

11. Mary bought a square painting of area $625$ cm². What is the amount of wood required to make the frame?

12. A school auditorium has $4624$ total seats if the number of seats in the row is equal to the number of seats in the columns. Which of the following shows the total number of seats in a row?

13. The formula for calculating the speed of a wave is given as $V=\sqrt{9.8d}$, where $d$ is the ocean’s depth in meters. What is the speed of the wave when the depth is $1500$ meters?

14. Which of the following shows the simplified expression for $2 + 9 – \sqrt{15}−2$?

15. Which of the following shows the simplified expression for $3\times 4 + \sqrt{169}$?

16. Which of the following shows the simplified expression for $\sqrt{25} \times \sqrt{16} + \sqrt{36}$?

17. Which of the following shows the simplified expression for $\sqrt{81} \times 12 + 12$?

18. Which of the following shows the simplified expression for $\sqrt{36} + \sqrt{47} – \sqrt{16}$?

19. Which of the following shows the simplified expression for $6 + \sqrt{36} + 25−2$?

20. Which of the following shows the simplified expression for $4(5) + \sqrt{9} − 2$?

21. Which of the following shows the simplified expression for $15 + \sqrt{16} + 5$?

22. Which of the following shows the simplified expression for $3\left(2\right)+\sqrt{25}+10$?

23. Which of the following shows the simplified expression for $4(7) + \sqrt{49} − 12$?

24. Which of the following shows the simplified expression for $2(4) + \sqrt{9} − 8$?

25. Which of the following shows the simplified expression for $3(7) + \sqrt{25} + 21$?

26. Which of the following shows the simplified expression for $8(3) – \sqrt{27}$?

27. What is the area of a right triangle with a hypotenuse of length $100$ cm and $60$ cm width?

28. What is the area of a right triangle with a hypotenuse of length $130$ cm and $50$ cm width?