Powers of Powers

Algebra 1 · Axiom Academy

One bacterium splits into two, and two into four, and four into eight. Watch that doubling explode — then discover the shortcut for stacking the growth. Why repeated doubling gets out of hand fast A single bacterium reproduces by splitting in two. Every generation, every cell splits, so the whole population doubles — again and again. After only a handful of generations the count is enormous, and writing it as 2 2 2 gets unwieldy. Exponents are how we keep that growth under control. Press play and watch one cell split, then split again, then again. Each generation every cell doubles, so the running count climbs 1,\ 2,\ 4,\ 8,\ 16, — that is 2^ g after g generations. Doubling each step is a power of two: the exponent just counts the generations. Drive the generations yourself Slide the dial to choose how many generations pass. The colony fills in, and the readout shows the population as 2^ n . Notice how a few extra generations multiply the count many times over. Each notch to the right doubles the count — that runaway climb is exponential growth. Stacking the growth: a power of a power Suppose the colony runs a generations, and you repeat that whole run b times in a row. Each run multiplies the count by 2^ a , and you do it b times. Drag the two dials and watch the exponents multiply : (2^ a )^ b = 2^ a b . All those doublings line up end to end: the total number of generations is just a b . One rule for every power of a power

This is the written version of the interactive lesson above. See the full Algebra 1 course.