AI Tutor
If an investment grows by 15%15\% in one year, does it grow by 15Γ·12=1.25%15 \div 12 = 1.25\% each month? Not quite β€” because growth compounds on top of previous growth.
To find the true monthly rate, we use exponent properties to rewrite the expression into an equivalent form without changing its total value.

The Power of a Power Rule

When you raise a power to another power, you multiply the exponents: (am)n=amβ‹…n(a^m)^n = a^{m \cdot n}.
This works because repeated grouping is multiplication. For example, (x2)3(x^2)^3 means (xβ‹…x)β‹…(xβ‹…x)β‹…(xβ‹…x)(x \cdot x) \cdot (x \cdot x) \cdot (x \cdot x), which gives 66 factors of xx, or x6x^6.
What happens when we run this rule in reverse? That lets us break big growth rates down into smaller intervals.

Regrouping Exponents

Since mβ‹…nm \cdot n can be factored, any expression in the form bmβ‹…tb^{m \cdot t} can be rewritten as (bm)t(b^m)^t.
Consider 23t2^{3t}. We can group the 33 inside the base as (23)t(2^3)^t, which simplifies directly to 8t8^t.
A common mistake is confusing (bm)t=bmβ‹…t(b^m)^t = b^{m \cdot t} with bmβ‹…bt=bm+tb^m \cdot b^t = b^{m + t}. Remember: multiplying bases with powers adds exponents, but raising a power to a power multiplies them.
Integers make numbers grow faster, but what if we need to zoom in to a smaller unit of time, like months or days?