Question

these numbers are in order from largest to smallest.
$4.2 \cdot 10^{-2}$
$.007$
$6.5 \cdot 10^{-3}$
\bigcirc true
\bigcirc false

Explanation:

Step1: Convert to decimal

Convert \(4.2 \cdot 10^{-2}\) to decimal: \(4.2 \cdot 10^{-2}=\frac{4.2}{100} = 0.042\).
Convert \(6.5 \cdot 10^{-3}\) to decimal: \(6.5 \cdot 10^{-3}=\frac{6.5}{1000}=0.0065\).
The number \(.007\) is \(0.007\).

Step2: Compare values

Now we have the numbers: \(0.042\) (\(4.2 \cdot 10^{-2}\)), \(0.007\) (\(.007\)), and \(0.0065\) (\(6.5 \cdot 10^{-3}\)).
Compare them: \(0.042>0.007>0.0065\)? Wait, \(0.007 = 0.007\) and \(0.0065=0.0065\), so \(0.007>0.0065\), but \(0.042>0.007\). Wait, no, wait: \(0.042\) (4.2e-2) is 0.042, \(0.007\) is 0.007, \(0.0065\) is 0.0065. So order from largest to smallest should be \(4.2 \cdot 10^{-2}\) (0.042), then \(.007\) (0.007), then \(6.5 \cdot 10^{-3}\) (0.0065)? Wait, but \(0.007\) is 0.007, and \(6.5 \cdot 10^{-3}\) is 0.0065, so \(0.007>0.0065\), and \(4.2 \cdot 10^{-2}=0.042>0.007\). So the given order is \(4.2 \cdot 10^{-2}\), \(.007\), \(6.5 \cdot 10^{-3}\). Let's check the values: \(0.042\) (first), \(0.007\) (second), \(0.0065\) (third). So \(0.042>0.007>0.0065\)? Wait, \(0.007\) is 0.007, \(0.0065\) is 0.0065, so \(0.007>0.0065\), and \(0.042>0.007\). So the order from largest to smallest is \(4.2 \cdot 10^{-2}\) (0.042), then \(.007\) (0.007), then \(6.5 \cdot 10^{-3}\) (0.0065). Wait, but wait: \(0.007\) is 0.007, which is 7 thousandths, and \(6.5 \cdot 10^{-3}\) is 6.5 thousandths, so 7 thousandths is more than 6.5 thousandths. And 4.2 hundredths is 42 thousandths, which is more than 7 thousandths. So the order given is largest to smallest: \(4.2 \cdot 10^{-2}\), \(.007\), \(6.5 \cdot 10^{-3}\). Which is correct? Wait, \(0.042>0.007>0.0065\), so yes, that's correct? Wait no, wait: \(0.042\) is 42 thousandths, \(0.007\) is 7 thousandths, \(0.0065\) is 6.5 thousandths. So 42 thousandths > 7 thousandths > 6.5 thousandths. So the order is correct? Wait, but wait, maybe I made a mistake. Wait, \(4.2 \cdot 10^{-2}\) is 0.042, \(.007\) is 0.007, \(6.5 \cdot 10^{-3}\) is 0.0065. So 0.042 > 0.007 > 0.0065. So the order from largest to smallest is correct. Wait, but wait, the user's question is whether the numbers are in order from largest to smallest. So according to the conversion, \(4.2 \cdot 10^{-2}=0.042\), \(.007 = 0.007\), \(6.5 \cdot 10^{-3}=0.0065\). So 0.042 > 0.007 > 0.0065, so the order is correct. Wait, but wait, maybe I messed up the decimal places. Let's recheck: \(10^{-2}\) is hundredth, so \(4.2 \times 10^{-2}=4.2/100 = 0.042\) (correct). \(10^{-3}\) is thousandth, so \(6.5 \times 10^{-3}=6.5/1000 = 0.0065\) (correct). \(.007\) is 0.007 (which is 7/1000 = 0.007). So 0.042 (42/1000) vs 0.007 (7/1000) vs 0.0065 (6.5/1000). So 42/1000 > 7/1000 > 6.5/1000. So yes, the order is largest to smallest. Wait, but the options are True or False. So is the answer True? Wait, no, wait: wait, 0.007 is 0.007, which is 7 thousandths, and 6.5e-3 is 6.5 thousandths, so 7 thousandths is more than 6.5 thousandths. And 4.2e-2 is 42 thousandths, which is more than 7 thousandths. So the order is 42 thousandths, 7 thousandths, 6.5 thousandths, which is largest to smallest. So the statement is True? Wait, but maybe I made a mistake. Wait, let's write all as thousandths: 4.2e-2 = 42/1000, 0.007 = 7/1000, 6.5e-3 = 6.5/1000. So 42/1000 > 7/1000 > 6.5/1000. So yes, the order is from largest to smallest. So the answer is True? Wait, but the original problem: let me check again. The numbers are \(4.2 \cdot 10^{-2}\), \(.007\), \(6.5 \cdot 10^{-3}\). So converted to decimals: 0.042, 0.007, 0.0065. So 0.042 > 0.007 > 0.0065. So the order is largest to smallest. So the statement is True. Wait, but maybe I messed up. Wait, 0.007 is 0.007, which is 7 thousandths, and 6.5e-3 is 6.5 thousandths, so 7 > 6.5, so 0.007 > 0.0065. And 0.042 is 42 thousandths, which is more than 7 thousandths. So yes, the order is correct. So the answer is True? Wait, but the options are True or False. So the correct answer is True? Wait, no, wait: wait, 0.007 is 0.007, which is 7/1000, and 6.5e-3 is 6.5/1000, so 7/1000 > 6.5/1000. And 4.2e-2 is 42/1000, which is > 7/1000. So the order is 42/1000, 7/1000, 6.5/1000, which is largest to smallest. So the statement is True. So the answer is True. Wait, but maybe I made a mistake. Let me check with another approach. Let's write all numbers in scientific notation with the same exponent. Let's use \(10^{-3}\). So \(4.2 \cdot 10^{-2}=42 \cdot 10^{-3}\) (because \(10^{-2}=10 \cdot 10^{-3}\), so 4.2*10=42). \(.007 = 7 \cdot 10^{-3}\) (because 0.007 = 7/1000 = 7*10^{-3}). \(6.5 \cdot 10^{-3}\) is already \(6.5 \cdot 10^{-3}\). So now we have \(42 \cdot 10^{-3}\), \(7 \cdot 10^{-3}\), \(6.5 \cdot 10^{-3}\). Now, comparing the coefficients: 42 > 7 > 6.5. So the numbers in scientific notation with the same exponent are in order from largest to smallest (since 42*10^-3 >7*10^-3 >6.5*10^-3). Therefore, the original numbers (4.2*10^-2, 0.007, 6.5*10^-3) are in order from largest to smallest. So the statement is True.

Answer:

True