Understanding Exponents, Divisibility, and Fractions

Posted on Feb 6, 2025 in Statistics

Understanding Exponents

Exponentiation is a multiplication of equal factors. For example: 2.2.2.2.2 = 25 (where 5 is the exponent and 2 is the base). The base indicates how many times it is multiplied by itself.

  • 43 = 4.4.4 = 64
  • 8.8.8.8.8 = 85
  • Base 2, exponent 6 = 64
  • Base 0, exponent 9 = 0

Order of operations with exponents:

  1. Powers
  2. Multiplications
  3. Additions and subtractions

For example: 3.24 + 25 = 3.16 (= 24) + 32 (= 25) = 48 + 32 = 80.

Properties of Exponentiation

  • Multiplication of powers with the same base: 5.5.5 = 51+1+1 = 53 (Keep the base and add the exponents).
  • Division of powers with the same base: 26 : 22 = 26-2 = 24 (Keep the base and subtract the exponents).
  • Power of a power: (25)3 = 25.3 = 215 (Keep the base and multiply the exponents).

Special Cases

  • Any power with an exponent equal to 1 is equal to the base: 21 = 2
  • Any power with an exponent of 0 is equal to 1: 20 = 1

Simplifying Powers

104 . 103 = 10.10.10.10 x 10.10.10 = 107

Simplifying an expression means transforming it into an expression with fewer operations but the same result.

Divisibility and Multiples of Natural Numbers

Divisibility: A natural number is divisible by another when the division of the first by the second is exact. For example, 427 is divisible by 7 because there is no remainder.

Divisibility Criteria

  • By 2: All even numbers are divisible by 2.
  • By 3: A number is divisible by 3 when the sum of its digits is divisible by 3.
  • By 5: All numbers ending in 0 or 5 are divisible by 5.
  • By 6: All numbers divisible by both 2 and 3 are divisible by 6.
  • By 4: When the last two digits form a number divisible by 4.
  • By 10: All numbers ending in 0 are divisible by 10.
  • By 8: When the last three digits form a number divisible by 8.
  • By 9: When the sum of its digits is divisible by 9.

Prime Numbers

Prime numbers up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Decomposition into Prime Factors

To decompose a number into prime factors, we indicate the number as a multiplication of prime numbers. For example, 60 = 2.2.3.5.

Factoring: To factor a composite number is to decompose it into a product of prime factors. For example: 48 = 24.3.

Multiples

Finding the multiples of a number involves multiplying that number by the natural numbers. For example, the first 5 multiples of 7 are 0, 7, 14, 21, and 28.

MMC (Least Common Multiple)

To find the MMC of multiple numbers, start by dividing by prime numbers. It is not necessary for all numbers to be divisible, but the goal is to reach 1 for all. Then, multiply the prime numbers used.

MDC (Greatest Common Divisor)

The MDC of two or more natural numbers is the largest number that divides all of them. Divide both numbers by the same prime number. If it is no longer possible to divide by the same number, multiply the divisors used.

Fractions

A fraction represents a part of a whole. For example, 5/8 (where 5 is the numerator and 8 is the denominator).

Types of Fractions

  • Proper: The numerator is smaller than the denominator.
  • Improper: The numerator is greater than or equal to the denominator.
  • Apparent: The numerator is a multiple of the denominator.

Mixed Numbers

Any improper or apparent fraction can be written as a mixed number, which consists of a whole number and a fractional part. To convert a mixed number to an improper fraction, multiply the denominator by the whole number and add the numerator. For example, 2 3/4 = 11/4.

Equivalent Fractions

Two or more fractions that represent the same quantity are called equivalent fractions. To check, multiply the numerator of one fraction by the denominator of the other. If the results are equal, the fractions are equivalent.

Simplifying Fractions

To simplify a fraction, divide the numerator and denominator by a common divisor. For example: 24/36 : 2 = 12/18 : 2 = 6/9 : 3 = 2/3.

Reducing Fractions to the Same Denominator

Calculate the MMC of the denominators. This MMC will be the new common denominator. Divide the common denominator by the original denominator of each fraction and multiply the result by the numerator.

Comparing Fractions

  • Equal denominators: The fraction with the smaller numerator is smaller.
  • Different numerators and denominators: Reduce the fractions to the same denominator first.

Operations with Fractions

  • Addition and subtraction with equal denominators: Keep the denominator and add or subtract the numerators.
  • Addition and subtraction with different denominators: Reduce to the same denominator, then divide the new denominator by the old denominator, and multiply the result by the old numerator. The result will be the new numerator.
  • Multiplication of fractions: Multiply the numerators together and the denominators together.