Consider a number. Greater than 2 which do you think is larger, the sum if you add a number to itself 10 times, or the product if you multiply the number by itself 10 times? Explain

Exponential growth(n^10) is greater.

To compare the sum of adding a number to itself 10 times versus the product of multiplying the number by itself 10 times, we can start by understanding the process for each operation.

Adding a number to itself 10 times:
If we have a number 'x', we can add it to itself 10 times by simply multiplying 'x' by 10:

Sum = x + x + x + x + x + x + x + x + x + x = 10x

Multiplying the number by itself 10 times:
If we multiply a number 'x' by itself 10 times, we are essentially raising 'x' to the power of 10:

Product = x * x * x * x * x * x * x * x * x * x = x^10

The question is whether the sum of 10x is larger or the product of x^10 is larger.

To determine this, we need to compare the values of these expressions. Let's consider an example with a number like 3:

For the sum:
Sum = 10 * 3 = 30

For the product:
Product = 3^10 = 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 = 59049

From this example, we can see that the product (59049) is significantly larger than the sum (30).

In general, when comparing the sum of adding a number to itself multiple times versus the product of multiplying the number by itself multiple times, the product tends to be larger, especially as the number of repetitions increases. This is because exponentiation (raising a number to a power) has a compounding effect, causing the product to increase rapidly.

Therefore, in the case of comparing the sum and product when adding or multiplying a number by itself multiple times, the product is usually larger.