Powers

Definition of Powers

• A power is a number expressed using two numbers, a base and an exponent.
• The base is the number that is being multiplied and the exponent refers to how many times the base number is being multiplied by itself.
• If you have 2^3, 2 is the base and 3 is the exponent. This equates to 222 = 8.

Calculating Powers

• Squaring a number simply means multiplying the number by itself. For example, 4^2 is 4 * 4 which equals 16.
• Cubing a number means multiplying a number by itself twice. For instance, 3^3 equals 3 * 3 * 3, resulting in 27.
• Powers can also be displayed using a negative exponent. For example, 2^-2 equals 1/(2^2) = 1/4.

Using Powers in Arithmetic

• When you multiply powers with the same base you add the exponents. For instance, 2^3 * 2^2 equals 2^(3+2) or 2^5.
• When you divide powers with the same base you subtract the exponents. For example, 3^4 / 3^2 equals 3^(4-2) or 3^2.
• When a power is raised to another power, you multiply the exponents. For example, (3^2)^3 is the same as 3^(2*3) or 3^6.

Application of Powers

• Powers are widely used in all areas of mathematics and science, including calculations around areas and volumes, exponential growth and decay, and statistical analyses.
• Square units (like m^2) might represent area, and cubic units (like m^3) might denote volume.

Common Mistakes with Powers

• Misplacing the base and exponent is a common mistake. Always remember the base is the number being multiplied and the exponent is the multiplier.
• Neglecting to follow the correct order of operations when dealing with exponents and other operations can lead to errors.
• Forgetting to apply exponent rules correctly, such as when you are multiplying, dividing or raising one power to another, can also result in mistakes.
• Neglecting the rule of any non-zero number to the zero power equals to 1. For example, 5^0 = 1.