An exponent shows repeated multiplication. Two cubed means two times two times two equals eight. Three squared means three times three equals nine. Five cubed is one hundred twenty five. Ten to the fourth power is ten thousand.
The product rule says that when multiplying powers with the same base, we add the exponents. Two cubed times two squared equals two to the fifth which is thirty two. Similarly three squared times three to the fourth equals three to the sixth which is seven hundred twenty nine. In algebra, x cubed times x to the fifth equals x to the eighth.
The quotient rule says that when dividing powers with the same base, subtract the exponents. Two to the fifth divided by two squared equals two to the third or eight. In algebra, x to the seventh divided by x to the fourth equals x cubed.
The power rule states that raising a power to another power multiplies the exponents. Two cubed squared equals two to the sixth or sixty four. The product to a power rule says that a product raised to an exponent applies to each factor. For example, three times four squared equals nine times sixteen or one hundred forty four. Similarly, a quotient raised to a power applies to numerator and denominator.
Any non zero number to the power of zero equals one. This follows from the quotient rule such as two cubed divided by two cubed equals two to the zero which is one. A negative exponent means the reciprocal of the base to the positive exponent. For example, two to the minus three equals one over two cubed which is one eighth.
We can combine multiple exponent laws. For example, two cubed times two to the negative one equals two squared, squared is two to the fourth, divided by two to the fourth is one. Another example, x squared y cubed to the power of minus one is x to the minus two y to the minus three, times x y to the fourth equals x squared y to the first.