Index Laws with Integer Exponents
What an exponent counts
Exponents are mathematics' way of writing repeated multiplication without the cramp. Instead of 2 × 2 × 2 × 2 × 2, we write 25, 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. 25 is 32. 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. 23 × 24 = 27, which is 128; you can check by hand that 8 × 16 really is 128. The reason is just careful counting: three copies of 2, then four more copies of 2, is seven copies in total. The same logic runs on letters, so x5 × x3 = x8. Notice the law only applies when the bases match; it says nothing about 23 × 33.
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. 27 ÷ 24 = 23, or 8, which matches 128 ÷ 16. 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. (23)2 = 26, which is 64, the same as 82. 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. (32)3 = 36, which is 729, not 35.
Why anything to the power zero is one
The quotient law forces a beautiful question. What is 23 ÷ 23? Subtracting the exponents gives 20, yet any number divided by itself is 1. So 20 must be 1, and the same argument makes any non-zero base to the power zero equal to 1. 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: 50 = 1, and 50 + 70 = 2, not 0.
Negative exponents mean reciprocals
Push the same idea one step further and negative exponents appear. 20÷ 24, by the quotient law, is 2−4; but 20 is 1, so it is also 1/24. So a negative exponent means a reciprocal: 2−3 = 1/23, which is 1/8, and 3−2 = 1/9. The minus sign does not make the number negative; it flips it underneath a 1. 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. (2x2)3 raises the 2 to the third, giving 8, and raises x2 to the third, giving x6, for a tidy 8x6. In the same way, (a2b3)2 becomes a4b6, each exponent doubled. Combining laws in one line, x3 × x−5 = x−2, which is 1/x2.