Operations with Integers, Rational Numbers and Exponents

Integers are whole numbers and their opposites (negative numbers), including zero. They do not include fractions or decimals. Operations with integers follow specific rules for signs:\n- Addition: If signs are the same, add the absolute values and keep the sign. If signs are different, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.\n- Subtraction: Change the subtraction to addition of the opposite number (e.g., a - b = a + (-b)).\n- Multiplication and Division: If signs are the same, the answer is positive. If signs are different, the answer is negative.

Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. This includes integers, common fractions, mixed numbers, terminating decimals, and recurring decimals. Operations with rational numbers:\n- Addition and Subtraction of Fractions: Find a common denominator, then add or subtract the numerators.\n- Multiplication of Fractions: Multiply the numerators and multiply the denominators.\n- Division of Fractions: Multiply the first fraction by the reciprocal of the second fraction.\n- Operations with Decimals: Align decimal points for addition/subtraction. For multiplication, multiply as whole numbers then count decimal places. For division, make the divisor a whole number by shifting decimal points.

An exponent (or index) is a small number written above and to the right of a base number, indicating how many times the base number is multiplied by itself. For example, in a^n, 'a' is the base and 'n' is the exponent. The entire expression 'a^n' is called a power.\n\nLaws of Exponents:\n1. Product Rule: a^m × a^n = a^(m+n)\n2. Quotient Rule: a^m ÷ a^n = a^(m-n) (where a ≠ 0)\n3. Power Rule: (a^m)^n = a^(m×n)\n4. Zero Exponent: a^0 = 1 (where a ≠ 0)\n5. Negative Exponent: a^(-n) = 1/a^n (where a ≠ 0)\n6. Power of a Product: (ab)^n = a^n × b^n\n7. Power of a Quotient: (a/b)^n = a^n / b^n (where b ≠ 0)

When an expression involves multiple operations, we follow a specific order to ensure a consistent result. The acronym BODMAS helps us remember this order:\nB - Brackets (Parentheses)\nO - Orders (Exponents or Powers)\nD - Division and M - Multiplication (from left to right, whichever comes first)\nA - Addition and S - Subtraction (from left to right, whichever comes first)