Exponents, Scientific Notation, and Irrational Numbers

An exponent tells us how many times a base number is multiplied by itself. For example, in 'a^n', 'a' is the base and 'n' is the exponent. Integer exponents include positive whole numbers, zero, and negative whole numbers. Understanding the properties of exponents allows us to simplify expressions efficiently.\n\nProperties of Integer Exponents:\n1. Product of Powers: When multiplying powers with the same base, add the exponents.\n2. Quotient of Powers: When dividing powers with the same base, subtract the exponents.\n3. Power of a Power: When raising a power to another power, multiply the exponents.\n4. Power of a Product: To raise a product to a power, raise each factor to the power.\n5. Power of a Quotient: To raise a quotient to a power, raise both the numerator and the denominator to the power.\n6. Zero Exponent: Any non-zero base raised to the power of zero is 1.\n7. Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive exponent.

Scientific notation is a way to write very large or very small numbers concisely. It is expressed as a product of two factors: a coefficient and a power of 10. The coefficient must be a number greater than or equal to 1 and less than 10 (1 ≤ M < 10). The power of 10 indicates how many places the decimal point was moved and in which direction (positive for large numbers, negative for small numbers).

Performing operations with numbers in scientific notation involves applying exponent rules and careful handling of the coefficients.\n\nMultiplication: Multiply the coefficients and add the exponents of 10.\nDivision: Divide the coefficients and subtract the exponents of 10.\nAddition/Subtraction: Adjust the numbers so they have the same power of 10, then add or subtract the coefficients. The power of 10 remains the same. After any operation, ensure the result is in proper scientific notation (1 ≤ M < 10).

Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Their decimal representations either terminate (e.g., 0.5) or repeat (e.g., 0.333...). \n\nIrrational numbers are numbers that cannot be expressed as a simple fraction p/q. Their decimal representations are non-terminating AND non-repeating. Famous examples include pi (π) and the square root of any non-perfect square (e.g., √2, √7).

Since irrational numbers have non-terminating, non-repeating decimals, we often need to approximate their values. For square roots, we can estimate by finding the two consecutive perfect squares that the number under the radical falls between. This helps us place the irrational number on a number line between two integers or estimate its decimal value.