Question

1. what is the value of n in the equation 2.6 × 10⁻² = (5.2 × 10⁷) ÷ (2 × 10ⁿ)?
2. simplify (14.1 × 10⁵) − (2.9 × 10⁵). express your answer in scientific notation.
3. what is the mass of 75,000 oxygen molecules? express your answer in scientific notation.

mass of one molecule of oxygen = 5.3 × 10⁻²³ gram
(image of oxygen molecule)

1. communicate and justify your friend says that the quotient of 9.2 × 10⁸ and 4 × 10⁻³ is 2.3 × 10⁵. is this answer correct? explain.

Explanation:

Question 3

Step1: Rewrite the division as fraction

We have the equation \(2.6\times10^{-2}=\frac{5.2\times 10^{7}}{2\times 10^{n}}\)

Step2: Simplify the right - hand side's coefficient and exponent separately

First, simplify the coefficient: \(\frac{5.2}{2} = 2.6\)
Then, simplify the exponent part using the rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\), so \(\frac{10^{7}}{10^{n}}=10^{7 - n}\)
So the right - hand side becomes \(2.6\times10^{7 - n}\)

Step3: Equate the exponents

Now our equation is \(2.6\times10^{-2}=2.6\times10^{7 - n}\)
Since the coefficients are equal, we can equate the exponents: \(- 2=7 - n\)

Step4: Solve for n

Add \(n\) to both sides: \(n-2 = 7\)
Add 2 to both sides: \(n=7 + 2=9\)

Step1: Factor out the common power of 10

We have \((14.1\times10^{5})-(2.9\times10^{5})=(14.1 - 2.9)\times10^{5}\)

Step2: Subtract the coefficients

\(14.1-2.9 = 11.2\)
So we get \(11.2\times10^{5}\)

Step3: Convert to scientific notation

Scientific notation is in the form \(a\times10^{b}\) where \(1\leqslant a<10\). We rewrite \(11.2\times10^{5}\) as \(1.12\times10^{6}\) (because \(11.2 = 1.12\times10^{1}\), so \(11.2\times10^{5}=1.12\times10^{1}\times10^{5}=1.12\times10^{6}\))

Step1: Recall the formula for total mass

The total mass \(M\) of \(N\) molecules is given by \(M=N\times m\), where \(N\) is the number of molecules and \(m\) is the mass of one molecule.
Here, \(N = 75000=7.5\times10^{4}\) and \(m = 5.3\times10^{-23}\) grams.

Step2: Multiply the coefficients and the exponents

First, multiply the coefficients: \(7.5\times5.3=39.75\)
Then, multiply the exponents using the rule \(a^{m}\times a^{n}=a^{m + n}\): \(10^{4}\times10^{-23}=10^{4-23}=10^{-19}\)
So we have \(39.75\times10^{-19}\)

Step3: Convert to scientific notation

Rewrite \(39.75\times10^{-19}\) as \(3.975\times10^{1}\times10^{-19}=3.975\times10^{-18}\) grams.

Answer:

\(n = 9\)

Question 4