Question

1. use your rule to write $10^4 \cdot 10^6$ with a single exponent. what does this tell you about the value of $10^{10}$?
2. the state of georgia has roughly $10^7$ human residents. each human has roughly $10^{13}$ bacteria cells in his or her digestive tract. how many bacteria cells are there in the digestive tracts of all the humans in georgia?

Explanation:

Problem 2 (assuming the first solvable problem is about \(10^4 \cdot 10^6\))

Step1: Recall exponent rule for multiplication

When multiplying exponents with the same base, we use the rule \(a^m \cdot a^n = a^{m + n}\). Here, \(a = 10\), \(m = 4\), and \(n = 6\).
\(10^4 \cdot 10^6 = 10^{4 + 6}\)

Step2: Calculate the sum of exponents

\(4 + 6 = 10\), so \(10^4 \cdot 10^6 = 10^{10}\). This tells us that when multiplying powers of 10, we add the exponents, and \(10^{10}\) is 10 multiplied by itself 10 times (or 10,000,000,000).

Step1: Identify the values to multiply

We have the number of humans in Georgia as \(10^7\) and the number of bacteria per human as \(10^{13}\). To find the total number of bacteria, we multiply these two values. Using the exponent rule \(a^m \cdot a^n = a^{m + n}\) where \(a = 10\), \(m = 7\), and \(n = 13\).
\(10^7 \cdot 10^{13}=10^{7 + 13}\)

Step2: Calculate the sum of exponents

\(7+13 = 20\), so \(10^7 \cdot 10^{13}=10^{20}\)

Answer:

\(10^{10}\) (and it shows that for \(10^m \cdot 10^n\), we add exponents to get \(10^{m + n}\))

Problem 3