What is 100,000 times 1/100,000, using the standard method of multiplication? Write the answer as 10 to a power.
The Correct Answer and Explanation is:
Answer:
100,000×1100,000=1=100100,000 \times \frac{1}{100,000} = 1 = 10^0100,000×100,0001=1=100
Explanation
To solve the expression 100,000×1100,000100{,}000 \times \frac{1}{100{,}000}100,000×100,0001 using the standard method of multiplication, we treat both numbers as powers of 10 and apply the rules of exponents.
Step 1: Express the Numbers as Powers of 10
- 100,000100{,}000100,000 is a power of 10. Since 100,000=105100{,}000 = 10^5100,000=105
- 1100,000\frac{1}{100{,}000}100,0001 is the reciprocal of 10510^5105, which equals 1100,000=10−5\frac{1}{100{,}000} = 10^{-5}100,0001=10−5
Step 2: Multiply Using the Law of Exponents
Using the rule:10a×10b=10a+b10^a \times 10^b = 10^{a + b}10a×10b=10a+b
Apply this to the current expression:105×10−5=105+(−5)=10010^5 \times 10^{-5} = 10^{5 + (-5)} = 10^0105×10−5=105+(−5)=100
Step 3: Interpret the Result
100=110^0 = 1100=1
This is a fundamental rule in mathematics: Any nonzero number raised to the power of 0 equals 1.
Why This Works:
Multiplying a number by its reciprocal (or multiplicative inverse) always results in 1. Here, 100,000100{,}000100,000 and 1100,000\frac{1}{100{,}000}100,0001 are multiplicative inverses:100,000×1100,000=1100{,}000 \times \frac{1}{100{,}000} = 1100,000×100,0001=1
When we express the numbers as powers of 10, we use exponent rules to simplify the calculation. This method avoids large number manipulation and shows a deeper understanding of the structure of numbers.
Conclusion:
The product of 100,000×1100,000100{,}000 \times \frac{1}{100{,}000}100,000×100,0001 is 1, which is equal to 100\boxed{10^0}100. This demonstrates how exponent rules simplify otherwise large or complex calculations.
