Question

select all of the true statements.
6 × 10⁴ is 30 times as much as 2 × 10⁵.
6 × 10⁻³ is 0.15 times as much as 4 × 10⁻².
4 × 10² is 8,000 times as much as 5 × 10⁻².
7 × 10⁻⁵ is 0.01 times as much as 7 × 10⁻⁴.
submit

Explanation:

To determine which statements are true, we check each one by dividing the first number by the second number and see if it equals the given multiple.

Statement 1: \( 6 \times 10^4 \) is 30 times as much as \( 2 \times 10^5 \)

• Calculate \( \frac{6 \times 10^4}{2 \times 10^5} \)
• Using exponent rules \( \frac{a^m}{a^n}=a^{m - n} \) and \( \frac{6}{2} = 3 \), we get \( 3\times10^{4 - 5}=3\times10^{-1}=0.3

eq 30 \). So this statement is false.

Statement 2: \( 6 \times 10^{-3} \) is 0.15 times as much as \( 4 \times 10^{-2} \)

• Calculate \( \frac{6 \times 10^{-3}}{4 \times 10^{-2}} \)
• \( \frac{6}{4}=1.5 \) and \( 10^{-3-(-2)} = 10^{-1} \), so \( 1.5\times10^{-1}=0.15 \). This statement is true.

Statement 3: \( 4 \times 10^2 \) is 8,000 times as much as \( 5 \times 10^{-2} \)

• Calculate \( \frac{4 \times 10^2}{5 \times 10^{-2}} \)
• \( \frac{4}{5}=0.8 \) and \( 10^{2-(-2)}=10^4 \), so \( 0.8\times10^4 = 8000 \). This statement is true.

Statement 4: \( 7 \times 10^{-5} \) is 0.01 times as much as \( 7 \times 10^{-4} \)

• Calculate \( \frac{7 \times 10^{-5}}{7 \times 10^{-4}} \)
• \( \frac{7}{7}=1 \) and \( 10^{-5-(-4)}=10^{-1} \), so \( 1\times10^{-1}=0.1

eq0.01 \). Wait, no: Wait, \( \frac{7\times 10^{-5}}{7\times 10^{-4}}=\frac{10^{-5}}{10^{-4}} = 10^{-1}=0.1 \)? Wait, no, let's recalculate:

Wait, \( 7\times10^{-5}\div(7\times10^{-4})=\frac{7}{7}\times\frac{10^{-5}}{10^{-4}} = 1\times10^{-5 + 4}=1\times10^{-1}=0.1
eq0.01 \). Wait, but wait, maybe I made a mistake. Wait, \( 7\times10^{-4}\times0.01=7\times10^{-4}\times10^{-2}=7\times10^{-6}
eq7\times10^{-5} \). Wait, no, let's do it again:

Wait, \( \frac{7\times 10^{-5}}{7\times 10^{-4}}=\frac{10^{-5}}{10^{-4}}=10^{-1}=0.1 \), which is not 0.01. Wait, but maybe the original problem's check was wrong. Wait, no, let's check the third statement again:

\( \frac{4\times10^2}{5\times10^{-2}}=\frac{4}{5}\times10^{2 + 2}=\frac{4}{5}\times10^4 = 0.8\times10^4 = 8000 \), that's correct.

Wait, for statement 4: \( 7\times10^{-5} \) and \( 7\times10^{-4} \). Let's see \( 7\times10^{-4}\times0.01=7\times10^{-4}\times10^{-2}=7\times10^{-6}
eq7\times10^{-5} \). But \( 7\times10^{-5}=7\times10^{-4}\times\frac{1}{10}=7\times10^{-4}\times0.1 \), so it's 0.1 times, not 0.01. So statement 4 is false. Wait, but maybe I messed up.

Wait, let's re-express all:

1. \( \frac{6\times10^4}{2\times10^5}=\frac{6}{2}\times10^{4 - 5}=3\times10^{-1}=0.3

eq30 \) (false)

1. \( \frac{6\times10^{-3}}{4\times10^{-2}}=\frac{6}{4}\times10^{-3 + 2}=1.5\times10^{-1}=0.15 \) (true)
2. \( \frac{4\times10^2}{5\times10^{-2}}=\frac{4}{5}\times10^{2+2}=0.8\times10^4 = 8000 \) (true)
3. \( \frac{7\times10^{-5}}{7\times10^{-4}}=\frac{10^{-5}}{10^{-4}}=10^{-1}=0.1

eq0.01 \) (false)

So the true statements are the second and third ones.

Answer:

B. \( 6 \times 10^{-3} \) is 0.15 times as much as \( 4 \times 10^{-2} \)
C. \( 4 \times 10^2 \) is 8,000 times as much as \( 5 \times 10^{-2} \)