Index Laws with Integer Exponents
What an exponent counts
Exponents are mathematics' way of writing repeated multiplication without the cramp. Instead of two times two times two times two times two, we write two to the power of five, a small raised digit doing the work of four multiplication signs. The big number is the base, the little raised one is the exponent, and the exponent simply counts how many copies of the base are multiplied together. Two to the power of five is thirty-two. This unit takes that idea, which you already meet with whole numbers, and turns it into a set of laws that work for any integer exponent and, crucially, for variables too.
The product law: add the exponents
The first law is the product law. When you multiply two powers that share the same base, you keep the base and add the exponents. Two to the third times two to the fourth is two to the seventh, which is one hundred and twenty-eight; you can check by hand that eight times sixteen really is one hundred and twenty-eight. The reason is just careful counting: three copies of two, then four more copies of two, is seven copies in total. The same logic runs on letters, so x to the fifth times x to the third is x to the eighth. Notice the law only applies when the bases match; it says nothing about two to the third times three to the third.
The quotient law: subtract the exponents
The second law is the quotient law, the mirror image of the first. Dividing powers with the same base means you keep the base and subtract the exponents. Two to the seventh divided by two to the fourth is two to the third, or eight, which matches one hundred and twenty-eight divided by sixteen. Cancelling shared factors top and bottom is what the subtraction records.
The power law: multiply the exponents
The third law is the power law: raising a power to another power multiplies the exponents. Two to the third, all raised to the second power, is two to the sixth, which is sixty-four, the same as eight squared. Here you are making two copies of a block that already holds three twos, so six twos in all. Watch this one carefully, because multiplying the exponents is exactly where students who are rushing tend to add instead. Three squared, cubed, is three to the sixth, which is seven hundred and twenty-nine, not three to the fifth.
Why anything to the power zero is one
The quotient law forces a beautiful question. What is two to the third divided by two to the third? Subtracting the exponents gives two to the zero, yet any number divided by itself is one. So two to the zero must be one, and the same argument makes any non-zero base to the power zero equal to one. This is not an arbitrary convention; it is the only value that keeps the laws consistent. A common trap is to think a zero exponent gives zero, so it is worth saying plainly: five to the zero is one, and five to the zero plus seven to the zero is two, not zero.
Negative exponents mean reciprocals
Push the same idea one step further and negative exponents appear. Two to the power zero divided by two to the fourth, by the quotient law, is two to the power negative four; but two to the power zero is one, so it is also one over two to the fourth. So a negative exponent means a reciprocal: two to the negative three is one over two to the third, which is one eighth, and three to the negative two is one ninth. The minus sign does not make the number negative; it flips it underneath a one. This is the single most important idea to carry forward, because it lets every expression be rewritten with positive exponents whenever that is clearer.
Extending the laws to variables
Finally, these laws extend cleanly to variables wrapped in brackets, where a coefficient and several letters all feel the exponent at once. Two x squared, all cubed, raises the two to the third, giving eight, and raises x squared to the third, giving x to the sixth, for a tidy eight x to the sixth. In the same way, a squared times b cubed, all squared, becomes a to the fourth times b to the sixth, each exponent doubled. Combining laws in one line, x cubed times x to the negative five is x to the negative two, which is one over x squared.