Powers and Roots

Understanding Powers and Roots

  • Powers denote how many times a number, called the base, is multiplied by itself. For example, 2³ means 2 is multiplied by itself 3 times (2 x 2 x 2 = 8). Here, 2 is the base, and 3 is the power or exponent.

  • Roots are the reverse of powers. If 2 is the base and 3 is the power in 2³ = 8, then the ³√ of 8 is 2. This is also referred to as the cube root of 8.

  • The square root, cube root, and other higher roots indicate the number that would need to be multiply by itself to get the given number. For example, the square root of 16 is 4 (since 4 x 4 = 16), and the cube root of 27 is 3 (since 3 x 3 x 3 = 27).

Concepts of Powers and Roots

  • The power or exponent of a number tells you how many times to multiply the number by itself.

  • The power of 1 for any number is always the number itself. For example, 6¹ = 6.

  • Any number to the power of 0 equals 1. For instance, 6º= 1.

  • The most common roots are square and cube roots, corresponding to powers of 2 and 3, respectively.

  • A negative exponent means division, not multiplication. For example, 2⁻³ equals 1/(2³) or 1/8.

Practice Problems

  • Problem: Calculate the result of 3⁴.
    • Solution: 81
  • Problem: Identify the cube root of 64.
    • Solution: 4
  • Problem: Find the square root of 144.
    • Solution: 12

Key Points to Remember

  • Get comfortable with the concept of powers and how they are repeated multiplications of a number.

  • Understand that roots are the reverse of powers and signify the original number before the power operation.

  • Take note that the power of 1 and the power of 0 for any number are the number itself and 1, respectively.

  • Recognise that negative exponents imply division, not multiplication, and should be calculated accordingly.