**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.

**Example 1 : **

Solve the equation x² = 121 for "x".

**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

Hence, the solutions are 11 and −11.

**Example 2 : **

Solve the equation x² = 16/169 for "x".

**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

Hence, the solutions are 4/13 and −4/13.

**Example 3 : **

Solve the equation x³ = 729 for "x".

**Solution : **

x³ = 729

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

x = **∛**729

Apply the definition of cube root.

Think: What number cubed equals 729 ?

x = 9

Hence, the solutions is 9.

**Example 4 : **

Solve the equation x³ = 8/125 for "x".

**Solution : **

x³ = 8/125

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

x = **∛**(8/125)

Apply the definition of cube root.

Think: What number cubed equals 8/125 ?

x = 2/5

Hence, the solutions is 2/5.

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