Basics of Exponents Quiz

Definition: An exponent indicates how many times a number (the base) is multiplied by itself.
Notation: If ( a ) is the base and ( n ) is the exponent, it is written as ( a^n ).
Key properties:
- ( a^1 = a ): Any number to the power of one is itself.
- ( a^0 = 1 ): Any non-zero number raised to the power of zero is one.
- ( a^{-n} = \frac{1}{a^n} ): A negative exponent represents the reciprocal of the base raised to the positive exponent.
- ( a^m \times a^n = a^{m+n} ): When multiplying like bases, add the exponents.
- ( a^m \div a^n = a^{m-n} ): When dividing like bases, subtract the exponents.
- ( (a^m)^n = a^{m \cdot n} ): When raising a power to another power, multiply the exponents.
- ( a^m \times b^m = (a \times b)^m ): When multiplying different bases with the same exponent, combine the bases and keep the exponent.

Definition: Exponential equations involve variables in the exponent.
Common forms:
- ( a^x = b )
- ( a^x = a^y )
Methods of solution:
1. Same Base Method:
  - Rewrite both sides with the same base when possible.
  - Equate the exponents.
  - Example: ( 3^{2x} = 9 ) can be written as ( 3^{2x} = 3^2 ), thus ( 2x = 2 ) leading to ( x = 1 ).
2. Logarithm Method:
  - Use logarithms to solve ( a^x = b ).
  - Apply the logarithm to both sides: ( x \log(a) = \log(b) ).
  - Solve for ( x ): ( x = \frac{\log(b)}{\log(a)} ).
3. Graphical Method:
  - Graph both sides of the equation and find the intersection points.
4. Numerical Method:
  - Use numerical methods or calculators for complex equations that cannot be solved analytically.
Example: To solve ( 2^{3x} = 16 ):
- Rewrite ( 16 ) as ( 2^4 ).
- Set the exponents equal: ( 3x = 4 ).
- Solve for ( x ): ( x = \frac{4}{3} ).

An exponent signifies the number of times the base is multiplied by itself.
Exponential notation is expressed as ( a^n ), where ( a ) is the base and ( n ) is the exponent.
Key properties include:
- ( a^1 = a ): Any base raised to the power of one equals the base itself.
- ( a^0 = 1 ): Any non-zero base raised to zero is equal to one.
- ( a^{-n} = \frac{1}{a^n} ): Negative exponents represent the reciprocal of the base raised to the positive exponent.
- ( a^m \times a^n = a^{m+n} ): When multiplying like bases, the exponents are added.
- ( a^m \div a^n = a^{m-n} ): When dividing like bases, the exponents are subtracted.
- ( (a^m)^n = a^{m \cdot n} ): Raising a power to another power results in the exponents being multiplied.
- ( a^m \times b^m = (a \times b)^m ): When multiplying distinct bases with the same exponent, the bases are combined while keeping the exponent.

Exponential equations feature variables within the exponent.
Common forms of these equations include ( a^x = b ) and ( a^x = a^y ).
Solution methods:
- Same Base Method:
  - Rewrite both sides of the equation to have the same base, enabling exponent equating.
  - For example, ( 3^{2x} = 9 ) can be transformed to ( 3^{2x} = 3^2 ), leading to ( 2x = 2 ) and resulting in ( x = 1 ).
- Logarithm Method:
  - Logarithms are employed to resolve equations of the form ( a^x = b ).
  - Apply logarithms to both sides, yielding ( x \log(a) = \log(b) ) and then solve for ( x ) with ( x = \frac{\log(b)}{\log(a)} ).
- Graphical Method:
  - This method involves graphing both sides of the equation to find intersection points.
- Numerical Method:
  - For complex equations unsolvable analytically, numerical methods or calculators may be utilized.
Example illustration:
- To solve ( 2^{3x} = 16 ), rewrite ( 16 ) as ( 2^4 ) and set the exponents equal, resulting in ( 3x = 4 ), thus yielding ( x = \frac{4}{3} ).