Question

look at the following examples. can you figure out the pattern or rule being used in each one?
questions:

1. what patterns do you notice in each of the equations above?
2. can you describe a rule for each one—without using any math terms youve learned yet?

3^{2}cdot3^{4}=3^{6}
5^{7}div5^{3}=5^{4}
(2^{3})^{2}=2^{6}
4^{-2}=\frac{1}{4^{2}}

Explanation:

1. Patterns noticed: In the first equation, when multiplying numbers with the same base, the exponent of the result is the sum of the exponents of the numbers being multiplied. In the second equation, when dividing numbers with the same base, the exponent of the result is the difference of the exponents of the numbers being divided. In the third equation, when raising a power to another power, the exponent of the result is the product of the exponents. In the fourth equation, a negative - exponent means taking the reciprocal of the number with the positive version of that exponent.
2. Rules: For the first one, if you have two numbers with the same bottom number and they are being multiplied, the exponent of the answer is what you get when you add the exponents of the two numbers you multiplied. For the second, if you have two numbers with the same bottom number and they are being divided, the exponent of the answer is what you get when you subtract the exponent of the number you are dividing by from the exponent of the number you are dividing. For the third, if you have a number with an exponent and that whole thing has another exponent on it, the exponent of the answer is what you get when you multiply those two exponents. For the fourth, if a number has a negative exponent, you flip it (make it the bottom number of a fraction) and change the exponent to positive.

Answer:

1. Multiplication: sum of exponents for same - base numbers; Division: difference of exponents for same - base numbers; Power - of - a - power: product of exponents; Negative exponent: reciprocal with positive exponent.
2. Multiply same - base numbers: add exponents for answer; Divide same - base numbers: subtract exponents for answer; Raise a power to a power: multiply exponents for answer; Negative exponent: flip number and make exponent positive.