Simplify 2^3 * 2^2, and evaluate. Step 1: Expand both powers. 2^3 = 2*2*2 (three factors) and 2^2 = 2*2 (two factors). Step 2: Put them side by side: 2*2*2 * 2*2. Nothing has been multiplied together yet - it is one long string of factors of 2. Step 3: Count the factors: 3 + 2 = 5 factors. So 2^3 * 2^2 = 2^5, and 2^5 = 32. Check by values: 2^3 = 8 and 2^2 = 4, and 8 * 4 = 32. The law, stated: to multiply powers with the same base, keep the base and add the exponents - a^m * a^n = a^(m+n). The exponent is a factor count, and multiplying two strings of factors just joins the counts. Warning, said plainly: the law needs the same base. In 2^3 * 3^2 the factors are 2s and 3s; there is no single count to add, and no shortcut exists.

Simplify x^4 * x^3 -> x^7 (four factors of x joined with three more: 4 + 3 = 7 factors)