Final answer:
The student's question involves the exponent rule where x^p multiplied by x^q equals x to the power of (p+q). This is used to simplify expressions like 10^1 \cdot 10^3 which becomes 10^4, equaling 10,000.
Step-by-step explanation:
The student seems to be asking about a mathematical rule for exponents, specifically when multiplying numbers with the same base.
The rule to apply here is x^p \cdot x^q = x^{(p+q)}.
This rule means that when you multiply powers with the same base, you can add the exponents together.
For instance, when dealing with 3 squared (32) times 3 to the power of 5 (35), you would add the exponents together to get 3 to the power of 7 (37).
Similarly, 10 times 10 raised to the power of 3 is written as 101 \cdot 103, which, using our rule, becomes 10 to the power of 4 (104), because 1 + 3 = 4.
The result is 10,000.
Connecting this to the concept of integer powers, for example, 103 is simplifying 10 \cdot 10 \cdot 10, which equals 1,000.
On the other side, 10 to the power of negative 4 (10-4) is four repeated multiplications of 10 to the power of negative 1 (10-1), each representing 0.1, giving us a result of 0.0001.