Question

add and express the answer in scientific notation.
1
$(4.39 \times 10^4) + (2.71 \times 10^2)$
2
$(9.95 \times 10^5) + (8.96 \times 10^4)$
solve.
a red blood cell has a diameter of approximately $6.5 \times 10^{-3}$ mm. the width of a human hair is about 0.018 mm. which is wider (thicker)?

Explanation:

Problem 1: \((4.39 \times 10^{4}) + (2.71 \times 10^{2})\)

Step 1: Convert to same exponent

To add numbers in scientific notation, we first make the exponents the same. We can rewrite \(4.39\times 10^{4}\) as \(439\times 10^{2}\) (since \(10^{4}=10^{2 + 2}=10^{2}\times10^{2}\), so \(4.39\times 10^{4}=4.39\times10^{2}\times 10^{2}=439\times 10^{2}\)) and \(2.71\times 10^{2}\) remains as it is.
So the expression becomes \((439\times 10^{2})+(2.71\times 10^{2})\)

Step 2: Add the coefficients

Using the distributive property \(a\times c + b\times c=(a + b)\times c\), where \(a = 439\), \(b=2.71\) and \(c = 10^{2}\)
\((439 + 2.71)\times10^{2}=441.71\times 10^{2}\)

Step 3: Convert to scientific notation

Scientific notation is of the form \(a\times 10^{n}\) where \(1\leqslant a<10\) and \(n\) is an integer.
We rewrite \(441.71\times 10^{2}\) as \(4.4171\times 10^{2}\times10^{2}=4.4171\times 10^{4}\) (because \(441.71 = 4.4171\times100=4.4171\times 10^{2}\), and when we multiply \(10^{2}\times10^{2}=10^{4}\))

Step 1: Convert to same exponent

Rewrite \(9.95\times 10^{5}\) as \(99.5\times 10^{4}\) (since \(10^{5}=10^{4+1}=10^{4}\times10\), so \(9.95\times 10^{5}=9.95\times10\times 10^{4}=99.5\times 10^{4}\)) and \(8.96\times 10^{4}\) remains as it is.
The expression becomes \((99.5\times 10^{4})+(8.96\times 10^{4})\)

Step 2: Add the coefficients

Using the distributive property \((99.5 + 8.96)\times10^{4}=108.46\times 10^{4}\)

Step 3: Convert to scientific notation

Rewrite \(108.46\times 10^{4}\) as \(1.0846\times 10^{2}\times10^{4}=1.0846\times 10^{6}\) (because \(108.46=1.0846\times 100 = 1.0846\times 10^{2}\), and \(10^{2}\times10^{4}=10^{6}\))

Step 1: Convert \(0.018\) to scientific notation

\(0.018=\frac{18}{1000}=1.8\times 10^{-2}\) (because \(0.018 = 1.8\times10^{-2}\) since we move the decimal point 2 places to the right to get a number between 1 and 10)

Step 2: Compare the two numbers

We have the diameter of red blood cell \(d_{rbc}=6.5\times 10^{-3}\) and width of hair \(w_{hair}=1.8\times 10^{-2}\)
Let's convert them to the same exponent. \(1.8\times 10^{-2}=18\times 10^{-3}\)
Now compare \(6.5\times 10^{-3}\) and \(18\times 10^{-3}\). Since \(18>6.5\), \(18\times 10^{-3}>6.5\times 10^{-3}\)

Answer:

\(4.4171\times 10^{4}\)

Problem 2: \((9.95\times 10^{5})+(8.96\times 10^{4})\)