Final answer:
In an expression such as 6 to the 3rd power, the base is 6, and the exponent is 3. Multiplying 6 by itself three times equals 216. Familiarity with exponent rules enables manipulation of exponential terms in multiplication, division, and when raising an exponential term to another power.
Step-by-step explanation:
The relationship between the base and the exponents in an expression like 6 to the 3rd power is fundamental to understanding exponentiation in mathematics. In this case, 6 is the base and 3 is the exponent. The expression 6 to the 3rd power, denoted as 6³, instructs us to multiply the base, which is 6, by itself three times: 6 x 6 x 6, which equals 216.
The concept of exponentiation can be expanded when combining exponential terms. For instance, if we take the example of 3.2 × 10³ times 2 × 10², applying the rules of exponents, the result is 6.4 × 10⁵. Here, we see that when exponentiated terms are multiplied, the coefficients are multiplied separately, and the exponent terms are added together, as explained in the formula a⁴ × a² = a(4+2) = a⁶.
When an exponentiated term is raised to another power, such as (5³)4, the exponents are multiplied: 5(3×4) = 5ⁱ². Consequently, understanding the rules of exponents allows us to manipulate these numbers efficiently in various mathematical situations.