Index Laws with Numerical and Algebraic Bases

When you multiply powers of the same base, add the indices. Two cubed times two to the four means three twos multiplied by four more twos: seven twos altogether, so it is two to the seven. With numbers in front, multiply the numbers as usual and add the indices of each letter separately. Three a squared b times four a cubed b to the five: three times four is twelve, a squared times a cubed is a to the five, and b, which means b to the one, times b to the five is b to the six. The answer is twelve a to the five b to the six.

When you divide powers of the same base, subtract the indices. Five to the seven divided by five cubed: write it as a fraction, and the three fives on the bottom cancel three of the seven fives on top, leaving five to the four. Seven take away three is four. With numbers in front, divide the numbers and subtract the indices letter by letter. Twenty m to the eight n to the five, divided by four m cubed n squared: twenty divided by four is five, m to the eight take away three is m to the five, and n to the five take away two is n cubed. The answer is five m to the five n cubed.

A power of a power means repeat. x to the four, all cubed, is x to the four times x to the four times x to the four. Adding the indices gives four plus four plus four, which is twelve, and that is just four times three. So for a power of a power, multiply the indices: x to the four, all cubed, is x to the twelve. Compare that with x to the four times x cubed, with no brackets, which is x to the seven.

When a bracket holds a product, the power outside applies to every factor inside. Two a cubed, all to the four, is two to the four times a cubed to the four. Two to the four is sixteen, and a cubed to the four is a to the twelve, so the answer is sixteen a to the twelve. A fraction works the same way: x squared over three, all cubed, is x to the six over twenty-seven, because both the top and the bottom are cubed.

Anything divided by itself is one, so seven cubed divided by seven cubed is one. Subtract the indices instead and you get seven to the zero. Both ways must agree, so seven to the zero is one, and the same is true for any base except zero. Be careful with what the index touches. In five x to the zero, only the x is raised to zero, so it is five times one, which is five. In five x in brackets to the zero, the whole bracket is raised, so it is one. Quiz and exercises now.