Solve equations involving cubes and cube roots

Key Notes :

🟢 Solve Equations Involving Cubes and Cube Roots

Cube of a Number
  • The cube of a number a is a3=a×a×a
  • Example: 23=2×2×2=8
Cube Root of a Number
  • The cube root of a number b is a number x such that x3=b
  • Denoted as ∛b=x
  • Example: ∛27=3, because 33=27
Solving Equations Using Cubes
  • General form: x3=k
  • To solve: Take cube root on both sides:

x=∛k​

  • Example: x3=64

x=∛64=4

Solving Equations Using Cube Roots
  • General form: ∛x=k
  • To solve: Cube both sides:

x=k3

  • Example: ∛x=5

x=53=125

Equations with Constants on Both Sides

Example: x3+8=27

  1. Subtract 8 from both sides: x3 = 19
  2. Take cube root: x = ∛19
Negative Numbers
  • Cubes of negative numbers are negative:
    (−2)3=−8
  • Cube roots of negative numbers are negative:
    ∛−27=−3
Tips & Tricks
  • Only one real solution exists for cube and cube root equations.
  • Always isolate the cube or cube root first before solving.
  • Check your answer by cubing or taking cube root to verify.

💡 Quick Reminder:

∛x3 = x and (∛x)3 = x

Learn with an example

Look at this equation:

a3 =125

What is a, the cube root of 125?

a=___

You want to find the cube root of 125, so figure out which number cubed (multiplied by itself, and then multiplied by itself again) equals 125.

The number 5 cubed equals 125.

53 =5.5.5=125

In other words, the equation a3 =125 is true when a=5.

So the cube root of 125 is 5.

Look at this equation:

t3 =8

What is t, the cube root of 8?

t=____

You want to find the cube root of 8, so figure out which number cubed (multiplied by itself, and then multiplied by itself again) equals 8.

The number 2 cubed equals 8.

23 = 2.2.2 =8

In other words, the equation t3 =8 is true when t=2.

So the cube root of 8 is 2.

Look at this equation:

s3 = 27

What is s, the cube root of 27?

s=

You want to find the cube root of 27, so figure out which number cubed (multiplied by itself, and then multiplied by itself again) equals 27.

The number 3 cubed equals 27.

33=3.3.3=27

In other words, the equation 33=27 is true when s=3.

So the cube root of 27 is 3.

Let’s practice!