To determine which measurement of length is the smallest among the given measurements:
[tex]\[ 9 \times 10^{-6} \, \text{m} \][/tex]
[tex]\[ 7 \times 10^{-5} \, \text{m} \][/tex]
[tex]\[ 7 \times 10^{1} \, \text{m} \][/tex]
[tex]\[ 9 \times 10^{2} \, \text{m} \][/tex]
We'll compare their exponents and their coefficients.
1. Comparing Exponents: The negative exponents indicate very small numbers, as they represent powers of ten less than one. The positive exponents indicate larger numbers.
- [tex]\(9 \times 10^{-6} \, \text{m}\)[/tex] has an exponent of [tex]\(-6\)[/tex]
- [tex]\(7 \times 10^{-5} \, \text{m}\)[/tex] has an exponent of [tex]\(-5\)[/tex]
- [tex]\(7 \times 10^{1} \, \text{m}\)[/tex] has an exponent of [tex]\(1\)[/tex]
- [tex]\(9 \times 10^{2} \, \text{m}\)[/tex] has an exponent of [tex]\(2\)[/tex]
Since [tex]\(-6\)[/tex] is the smallest exponent, hence [tex]\(9 \times 10^{-6} \, \text{m} \)[/tex] will be the smallest in magnitude among the numbers in scientific notation.
2. Verifying and Concluding:
- [tex]\(9 \times 10^{-6} \, \text{m} = 0.000009 \, \text{m}\)[/tex]
- [tex]\(7 \times 10^{-5} \, \text{m} = 0.00007 \, \text{m}\)[/tex]
- [tex]\(7 \times 10^{1} \, \text{m} = 70 \, \text{m}\)[/tex]
- [tex]\(9 \times 10^{2} \, \text{m} = 900 \, \text{m}\)[/tex]
Clearly, [tex]\(0.000009 \, \text{m}\)[/tex] is the smallest value.
Thus, the smallest measurement of length is:
[tex]\[ 9 \times 10^{-6} \, \text{m} \][/tex]
