FINDING SQUARE ROOTS AND CUBE ROOTS

The square root of a positive number p is x if x² = p. There are two square roots for every positive number.

For example, the square roots of 36 are 6 and -6, because

6²  =  36 and (-6)²  =  36

The square roots of 1/25 are 1/5 and -1/5. We can write the square roots of 1/25 as ±1/5.

The symbol √ indicates the positive, or principal square root.

A number that is a perfect square has square roots that are integers. The number 81 is a perfect square because its square roots are 9 and -9.

The cube root of a positive number p is x, if x³ = p. There is one cube root for every positive number.

For example, the cube root of 8 is 2 because 2³ = 8.

The cube root of 1/27 is 1/3, because (1/3)³ = 1/27. 

The symbol √ indicates indicates the cube root.

A number that is a perfect cube has a cube root that is an integer.

The number 125 is a perfect cube because its cube root is 5.

Solved Examples

Example 1 : 

Solve for x :

x²  =  121

Solution : 

x²  =  121

Solve for x by taking the square root of both sides.

x  =  ± √121

Apply the definition of square root.

Think: What numbers squared equal 121 ?

x  =  ± 11

So, the solutions are 11 and −11.

Example 2 : 

Solve for x : 

x²  =  16/169

Solution : 

x²  =  16/169

Solve for x by taking the square root of both sides.

x  =  ± √(16/169)

Apply the definition of square root.

Think: What numbers squared equal 16/169 ?

x  =  ± 4/13

So, the solutions are 4/13 and −4/13.

Example 3 : 

Solve for x : 

x³  =  729

Solution : 

x³  =  729

Solve for x by taking the cube root of both sides.

x  =  ³√729

Apply the definition of cube root.

Think: What number cubed equals 729 ?

x  =  9

So, the solution is 9.

Example 4 : 

Solve for x :

x³  =  8/125

Solution : 

x³  =  8/125 

Solve for x by taking the cube root of both sides.

x  =   ³√(8/125)

Apply the definition of cube root.

Think: What number cubed equals 8/125 ?

x  =  2/5

So, the solution is 2/5.

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