To multiply numbers with exponents, if the base numbers are the same, keep the base number and add the exponents together. The general exponential notation is: x^a * x^b = x^(a + b), where "x" is the base, "a" and "b" are exponents. Let's look at an example: x^2 * x^3 = x^5. The key to understand it is to break this down to: x^2 * x^3 = (x * x) * (x * x * x). You can see "x" is written down two times (that is what x^2 means), then it is written three times (that is what x^3 means). So, "x" is written down 2 + 3 = 5 times. Put in the "times" as a symbol, you have x^5. The same rule applies when you have the same numbers as a base: 3^2 * 3^4 = 3^(2 + 4) = 3^6. The exponents "2" and "4" indicate the number of times to repeat the base "3" as a factor.
To take a power of a product of factors, raise each factor (the base) to that of the same power. The general exponential notation is: (x * y)^a = x^a * y^a. Note, the exponents are the same, but the bases are different. For example, (x * y)^2 = x^2 * y^2. Or--because all the laws of exponents work in both directions--x^2 * y^2 = (x * y)^2. To simplify this, break it down in terms of what exponents mean. "To the second" means "multiplying two times." Write ("expand") the two factors to the simplified form: (x * x) * (y * y) = (x * y) * (x * y). Multiplying the left side, will give you the same result as a product on the right.
To take a power of a power, multiply the exponents and applied to the base. The general exponential notation is: (x^a)^b = x^(a * b). For example: (x^2)^3 = x^(2 * 3). Let's break this down in terms of what exponents mean. Exponents ("to the third") mean that x^2 or (x * x) is multiplied by itself three times (x * x) * (x * x) * (x * x). So, you have three sets of (x * x) or a total of 6 "x" being multiplied. The same rule applies when the base is the number: (2^3)^4 = (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2). You have 12 two's being multiplied. So, (2^3)^4 = 2^(3 * 4) = 2^12.