A power is simplified way of expresing a product of the same factors.

For example, 3⁵ is a power. We read it "three to the power of five" and it means that we hav multiuplied the number 3 five times by itself: 3·3·3·3·3. The base is 3 (the number that will be repeated). And the exponent or index is 5 (the number of times that the base will be repeated).

The easiest powers are the power of base 1 or 10.

Special cases:

• 3² is read as:
  • Three to the second power
  • Three squared
  • Three to the power of two.
  • Three to the square power
  • The most common is: Three squared
• 5³ is read as:
  • Five cubed
  • Five to the third power
  • Five to the power of three.
  • The most common is: Five cubed

To manipulate expressions with powers we use some rules that are called laws of powers or laws of indices.

• Multiplication: When powers with the same base are multiplied, the base remains unchanged and the exponents are added: a^m·aⁿ = a^m+n . Example: 6⁷ · 6⁴ = 6⁷⁺⁴ =6¹¹
• Division: When powers with the same base are divided, the base remains unchanged and the exponents are subtracted: a^m:aⁿ = a^m-n . Example:

• Power of a Power: The exponents or indices must be multiplied: (aⁿ)^m= a^n·m. Example: (6⁷)⁴ = 6^7·4 = 6²⁸
• Multiplication of the same exponent: When powers with the same exponent are multiplied, multiply the bases and keep the same exponent: aⁿ · bⁿ= (a·b)ⁿ . Example: 2⁵·7⁵ = (2·7)⁵ = 14⁵.
• Divison of the same exponent: When powers with the same exponent are divided, bases are divided and the exponent remains unchanged: aⁿ : bⁿ = (a:b)ⁿ . Example: 6⁴ : 2⁴ = (6 : 2)⁴= 3⁴
• a⁰=1 (mathematical proof)
• a¹=a

The square root is the oposite operation to squared a number. For example, the square root of 16 is 4, because, 4²=16, and we write it . Where √ is called the radical sign or radix. The term whose root is being considered is known as the radicand.

√a = b means b² = a

There are numbers that are not squares of any other number, for example between 9 and 16 (squares of 3 and 4); there is not any whole square number.

The square root of all numbers between 9 and 16 are between 3 and 4, for example 3 < √13 < 4, this is an estimation of the √13 value.

When we estimate a square root as 3 < √13 < 4 we can also say that 13 is 3 and the difference of √13 and 9 (square of 3), which is 4, is called the remainder.

So we say that the square root of √13 is 9 and the remainder is 4. That means √13= 3² + 4.