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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.

How to Simplify Radicals?

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

Solve the following radical problem.

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. Write the following expressions in exponential form:

a) 7√y

b) 3√x 2

c) 6√ab

d)√w 2v 3

2. Simplify the following radicals.

a)3√x 8

b) √8y 3  

3. Simplify each of the following expressions.

a) √x (4 − 3√x)

b) (2√x + 1) (3 − 4√x)

4. A rectangular mat is 4 meters in length and √(x + 2) meters in width. Calculate the value of x if the perimeter is 24 meters.

5. 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. Calculate the total length of the spider web

6. Mary bought a square painting of area 625 cm 2. Calculate the amount of wood required to make the frame.

7. A kite is secured tied on the ground by a string. The wind blows such that the string is tight, and the kite is directly positioned on a 30 ft flag post. Find the height of the flag post if the length of the string is 110 ft long.

8. A school auditorium has 3136 total seats if the number of seats in the row is equal to the number of seats in the columns. Calculate the number total number of seats in a row.

9. The formula for calculating the speed of a wave is given as V=√9.8d, where d is the ocean’s depth in meters. Calculate the speed of the wave when the depth is 1500 meters.

10. A big squared playground is to be constructed in a city. If the playground area is 400 and is to be subdivided into four equal zones for different sporting activities. How many zones can be put in one row of the playground without surpassing it?

11. Simplify the following radical expressions:

  1. 2 + 9 –√15−2
  2. 3 x 4 + √169
  3. √25 x √16 + √36
  4. √81 x 12 + 12
  5. √36 + √47 – √16
  6. 6 + √36 + 25−2
  7. 4(5) + √9 − 2
  8. 15 + √16 + 5
  9. 3(2) + √25 + 10
  10. 4(7) + √49 − 12
  11. 2(4) + √9 − 8
  12. 3(7) + √25 + 21
  13. 8(3) – √27

12. Calculate the area of a right triangle that has a hypotenuse of length 100 cm and 6 cm width.

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