Question

question
what is the quotient of $2.592 \times 10^7$ and $7.2 \times 10^4$ expressed in scientific notation?
answer attempt 1 out of 2
answer: \\( \square \times 10^{\square} \\)

Explanation:

Step1: Divide the coefficients

To find the quotient of the two numbers in scientific notation, we first divide the coefficients. So we calculate $\frac{2.592}{7.2}$.
$2.592\div7.2 = 0.36$? Wait, no, let's do it properly. $2.592\div7.2$. Let's multiply numerator and denominator by 1000 to get rid of decimals: $\frac{2592}{7200}$. Simplify this fraction. Divide numerator and denominator by 48: $\frac{2592\div48}{7200\div48}=\frac{54}{150}$. Divide by 6: $\frac{9}{25} = 0.36$? Wait, no, wait $7.2\times0.36 = 2.592$? Let's check: $7\times0.36=2.52$, $0.2\times0.36 = 0.072$, so $2.52 + 0.072=2.592$. Yes, so the coefficient division gives $0.36$? Wait, no, wait, maybe I made a mistake. Wait, $2.592\div7.2$. Let's write it as $2.592\div7.2 = \frac{2.592}{7.2}$. Multiply numerator and denominator by 10: $\frac{25.92}{72}$. Now, $72\times0.36 = 25.92$ (since $72\times0.3 = 21.6$, $72\times0.06 = 4.32$, $21.6+4.32 = 25.92$). So the coefficient is $0.36$? Wait, but in scientific notation, the coefficient should be between 1 and 10. Oh, right! So we need to adjust that. Wait, no, wait, maybe I messed up the division. Wait, $2.592\times10^{7}\div(7.2\times10^{4})=(2.592\div7.2)\times(10^{7}\div10^{4})$.

Step2: Divide the powers of 10

Now, divide the powers of 10. Using the rule $a^{m}\div a^{n}=a^{m - n}$, so $10^{7}\div10^{4}=10^{7 - 4}=10^{3}$.

Step3: Adjust the coefficient to be in scientific notation

Now, the coefficient we got was $0.36$, but in scientific notation, the coefficient must be between 1 and 10 (excluding 10). So we can rewrite $0.36$ as $3.6\times10^{- 1}$. Then multiply by $10^{3}$: $3.6\times10^{-1}\times10^{3}=3.6\times10^{-1 + 3}=3.6\times10^{2}$. Wait, let's check again. Wait, $2.592\div7.2$: let's do it again. $7.2\times0.36 = 2.592$, but $0.36$ is less than 1, so we need to express it as $3.6\times10^{-1}$. Then $10^{7}\div10^{4}=10^{3}$. So multiplying the two parts: $(3.6\times10^{-1})\times10^{3}=3.6\times10^{2}$. Let's verify the original division: $(2.592\times10^{7})\div(7.2\times10^{4})=(2.592\div7.2)\times(10^{7}\div10^{4}) = 0.36\times10^{3}=3.6\times10^{2}$. Yes, that's correct. Because $0.36\times10^{3}=360$, and let's check $7.2\times10^{4}=72000$, $2.592\times10^{7}=25920000$. Now, $25920000\div72000 = 25920\div72 = 360$. And $3.6\times10^{2}=360$. So that's correct.

Answer:

$3.6\times10^{2}$