Question

question
what is the quotient of 3.013 × 10⁸ and 6.55 × 10⁵ expressed in scientific notation?
answer
answer: ×10
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Explanation:

Step1: Divide the coefficients

To find the quotient of the two numbers in scientific notation, we first divide the coefficients. So we calculate $\frac{3.013}{6.55}$.
\[ \frac{3.013}{6.55} = 0.46 \]
Wait, no, let's do the division properly. \(3.013\div6.55\):
\(6.55\times0.46 = 6.55\times(0.4 + 0.06)=6.55\times0.4+6.55\times0.06 = 2.62+0.393 = 3.013\). So \(\frac{3.013}{6.55}=0.46\)? Wait, no, that can't be. Wait, \(3.013\div6.55\): Let's do the division: \(3.013\div6.55\). Multiply numerator and denominator by 1000 to eliminate decimals: \(3013\div6550\). Let's divide 3013 by 6550. 6550 goes into 3013 zero times. Add a decimal point, 6550 goes into 30130 four times (4*6550 = 26200), subtract 26200 from 30130, we get 3930. Bring down a zero: 39300. 6550 goes into 39300 six times (6*6550 = 39300). So \(3.013\div6.55 = 0.46\)? Wait, no, 0.46*6.55 = 3.013, yes. Wait, but in scientific notation, the coefficient should be between 1 and 10. So we need to adjust. Wait, maybe I made a mistake. Wait, let's check again. Wait, \(3.013\times10^{8}\div(6.55\times10^{5})=(3.013\div6.55)\times10^{8 - 5}\).

Step2: Subtract the exponents

Now, for the exponents, when dividing numbers in scientific notation, we subtract the exponents of 10. So \(10^{8}\div10^{5}=10^{8 - 5}=10^{3}\).

Step3: Adjust the coefficient to be in scientific notation

Now, we have \(0.46\times10^{3}\). But the coefficient in scientific notation should be between 1 and 10. So we can rewrite \(0.46\) as \(4.6\times10^{- 1}\). Then, \(4.6\times10^{-1}\times10^{3}=4.6\times10^{-1 + 3}=4.6\times10^{2}\). Wait, wait, no, wait. Wait, \(3.013\div6.55\): Let's do this division again. \(3.013\div6.55\). Let's compute 3.013 ÷ 6.55. 6.55 × 0.46 = 3.013, as we saw. But 0.46 is less than 1, so we need to write it as 4.6 × 10⁻¹. Then multiply by 10³ (from the exponent subtraction: 8 - 5 = 3). So (4.6 × 10⁻¹) × 10³ = 4.6 × 10⁻¹⁺³ = 4.6 × 10². Wait, but let's check with a calculator. 3.013 ÷ 6.55 = 0.46? Wait, 6.55 × 0.4 = 2.62, 6.55 × 0.06 = 0.393, 2.62 + 0.393 = 3.013. Yes. So 0.46 × 10³. But 0.46 is 4.6 × 10⁻¹, so 4.6 × 10⁻¹ × 10³ = 4.6 × 10². So the coefficient is 4.6 and the exponent is 2. Wait, but let's check the calculation again. Wait, 3.013 × 10⁸ divided by 6.55 × 10⁵. So (3.013 / 6.55) × 10^(8 - 5). 3.013 / 6.55 = 0.46, then 0.46 × 10³. But 0.46 is 4.6 × 10⁻¹, so 4.6 × 10⁻¹ × 10³ = 4.6 × 10². So the quotient is 4.6 × 10². Wait, but let's verify with actual numbers. 3.013 × 10⁸ is 301300000, 6.55 × 10⁵ is 655000. Divide 301300000 by 655000. Let's divide numerator and denominator by 1000: 301300 / 655. 655 × 460 = 655 × 400 + 655 × 60 = 262000 + 39300 = 301300. So 301300 / 655 = 460. 460 in scientific notation is 4.6 × 10². Ah! There we go. I made a mistake earlier in the coefficient division. Wait, 3.013 × 10⁸ is 301300000, 6.55 × 10⁵ is 655000. 301300000 ÷ 655000. Let's cancel out three zeros from numerator and denominator: 301300 ÷ 655. 655 × 460 = 301300. So 460. 460 is 4.6 × 10². So where did I go wrong before? Oh, right! 3.013 × 10⁸ is 301300000, which is 3.013 × 10⁸, and 6.55 × 10⁵ is 655000. So 301300000 ÷ 655000. Let's divide numerator and denominator by 1000: 301300 ÷ 655. 655 × 460 = 301300. So 460. So 460 is 4.6 × 10². So the coefficient is 4.6 and the exponent is 2. So my initial division of 3.013 ÷ 6.55 was wrong. Wait, 3.013 × 10⁸ is 301300000, which is 3.013 × 10⁸, so 3.013 × 10⁸ ÷ 6.55 × 10⁵ = (3.013 ÷ 6.55) × 10^(8 - 5). 3.013 ÷ 6.55: Let's compute that. 6.55 × 0.46 = 3.013, but 3.013 × 10⁸ ÷ 6.55 × 10⁵ is (3.013 / 6.55) × 10^(3) = 0.46 × 10³ = 460, which is 4.6 × 10². Ah, right! Because 0.46 × 10³ = 460, and 460 is 4.6 × 10². So the correct coefficient is 4.6 and the exponent is 2.

Answer:

4.6 × 10²