Mastering Math Operations: Signs, Powers, and Roots

Posted on Feb 14, 2025 in Mathematics

Rules of Signs: Multiplication and Division

  • (+) × (+) = (+) Positive multiplied by positive equals positive.
  • (+) × (−) = (−) Positive multiplied by negative equals negative.
  • (−) × (+) = (−) Negative multiplied by positive equals negative.
  • (−) × (−) = (+) Negative multiplied by negative equals positive.
  • (+) ÷ (+) = (+) Positive divided by positive equals positive.
  • (+) ÷ (−) = (−) Positive divided by negative equals negative.
  • (−) ÷ (+) = (−) Negative divided by positive equals negative.
  • (−) ÷ (−) = (+) Negative divided by negative equals positive.

Working with Powers

Powers with the Same Base (Product)

When multiplying powers with the same base, add the exponents:

an ⋅ am = an+m

Example: 22 ⋅ 23 = 25

Powers with the Same Base (Quotient)

When dividing powers with the same base, subtract the exponents:

an ÷ am = an-m

Example: 25 ÷ 22 = 23

Product of Powers with Different Bases and Equal Exponents

When multiplying powers with different bases but the same exponent, multiply the bases and keep the exponent:

an ⋅ bn = (a ⋅ b)n

Example: 22 ⋅ 42 = (2 ⋅ 4)2

Quotient of Powers with Different Bases and Equal Exponents

When dividing powers with different bases but the same exponent, divide the bases and keep the exponent:

an ÷ bn = (a ÷ b)n

Example: 244 ÷ 34 = (24 ÷ 3)4

Power of a Power

When raising a power to another power, multiply the exponents:

(an)m = an⋅m

Example: (32)4 = 38

Exact and Inexact Roots

A root is exact when the result of raising it to the power of the index equals the radicand (the number under the radical). For example, 3 is the exact square root of 9 because 32 = 9. Similarly, 9 is the exact cube root of 729 because 93 = 729.

When there is no integer that, when raised to the power of the index, equals the radicand, the root is inexact or whole. For example, the square root of 38 is inexact because there is no integer that, when squared, equals 38. Inexact roots are called radicals.

Calculations with Roots

If you do not understand what a square root is, please review that concept first. All roots (square roots, cube roots, fourth roots, etc.) are estimated in the same way.

Adding and Subtracting Radicals

You cannot simplify expressions like √5 + √7 further. You can only approximate the value using a calculator. The same applies to subtraction: √3 − √2 can only be approximated.

However, you *can* add or subtract like radicals (radicals with the same radicand):

  • √5 + √5 = 2√5
  • 3√12 + √12 = 4√12
  • 6√20 + 10√20 − 3√20 = 13√20

And, of course, if you add or subtract *within* the root, you can calculate: √(15 + 19) = √34

Multiplying and Dividing Radicals

The rules for multiplication and division are different:

  • √a ⋅ √b = √ab
  • √a / √b = √(a/b)

Instead of multiplying the roots, you can multiply the radicands and place them under a single root. Instead of dividing the roots, you can divide the radicands and place them under a single root.

Examples:

  • √5 × √7 = √35
  • 0.1 × √10 = √1 = 1
  • √(1/4) × √32 = √8
  • √63 / √7 = √(63/7) = √9 = 3

These laws can also be used in reverse:

  • √150 = √(25 × 6) = √25 × √6 = 5√6
  • √(34/100) = √34 / √100 = √34 / 10

Combining Operations

Math exercises often combine many operations with roots. You need to be careful and practice!

  • √(5 × (3 + 17)) = √(5 × 20) = √100 = 10. First, add 3 and 17. Then, combine everything under one root (multiplying the radicands).
  • √((3 × 20) / 15) = √(60 / 15) = √4 = 2. First, multiply 3 by 20. Then, combine everything under one root.
  • √ (4 + 9) + 3√13 = √13 + 3√13= 4√13. First calculate the values inside the radicals, then add the like radicals.