\[\begin{align*} \boxed{10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000} \end{align*}\]
Introduction
Exponents provide a compact way to express large numbers through repeated multiplication. The expression \( 10^5 \) is an example of exponential notation, where:
– \( 10 \) is the base, representing the number being multiplied.
– \( 5 \) is the exponent, which tells us how many times to multiply the base by itself.
In other words, \( 10^5 \) means “multiply 10 by itself 5 times.”
Step-by-Step Solution
To solve \( 10^5 \), we can write out the multiplication:
\[\begin{align*} 10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000 \end{align*}\]
Base 10 Application
Each time we multiply by 10, we add another zero to the result. This is why \( 10^5 \) results in a 1 followed by 5 zeros. In general, \( 10^n \) equals 1 followed by \( n \) zeros, making powers of 10 very useful for representing large numbers. Base 10 exponentiation is regularly used as scientific notation in chemistry and physics!
Practice Problems
- Evaluate $10^3$.
Using the fact that powers of ten are just a $1$ followed by $n$ zeros, we have $10^3 = 1000$. - Evaluate $100^2$.
Just as powers of $10$ are followed by $n$ zeros, using $100$, it is followed by $2n$ zeros. This is easily seen by the fact that $100 = 10^2$. This means that $100^2 = (10^2)^2 = 10^{2 \cdot 2} = 10^4 = 10000$. - Evaluate $20^4$
Notice that $20 = 2 \cdot 10$ so this is the same as asking $2^4 \cdot 10^4$. Evaluating them separately, we get that $2^4 = 16$ and $10^4 = 10000$. Multiplying them, we get $16 \cdot 10000 = 160,000$.
