Question

operations with scientific notation
date
simplify. write each answer in scientific notation.

1. $(1.08 \times 10^{-1})(9.3 \times 10^{-3})$ 2) $(2 \times 10^{-4})(8.1 \times 10^{-1})$

1. $(2.32 \times 10^{-6})(4 \times 10^{-5})$ 4) $(3.48 \times 10^{3})(9.8 \times 10^{4})$

1. $(7.1 \times 10^{-5})(6.7 \times 10^{-6})$ 6) $(6 \times 10^{3})(9.91 \times 10^{0})$

1. $\frac{7.1 \times 10^{6}}{8.2 \times 10^{1}}$ 8) $\frac{5.4 \times 10^{-1}}{3.4 \times 10^{1}}$

Explanation:

Problem 1: \((1.08 \times 10^{-1})(9.3 \times 10^{-3})\)

Step 1: Multiply the coefficients

Multiply \(1.08\) and \(9.3\).
\(1.08 \times 9.3 = 10.044\)

Step 2: Add the exponents of 10

Add \(-1\) and \(-3\) (since \(10^a \times 10^b = 10^{a + b}\)).
\(-1 + (-3) = -4\)

Step 3: Adjust to scientific notation

The coefficient \(10.044\) is not between \(1\) and \(10\), so we rewrite it as \(1.0044 \times 10^1\). Then multiply by \(10^{-4}\):
\(1.0044 \times 10^1 \times 10^{-4} = 1.0044 \times 10^{-3}\)

Step 1: Multiply the coefficients

Multiply \(2\) and \(8.1\).
\(2 \times 8.1 = 16.2\)

Step 2: Add the exponents of 10

Add \(-4\) and \(-1\).
\(-4 + (-1) = -5\)

Step 3: Adjust to scientific notation

Rewrite \(16.2\) as \(1.62 \times 10^1\), then multiply by \(10^{-5}\):
\(1.62 \times 10^1 \times 10^{-5} = 1.62 \times 10^{-4}\)

Step 1: Multiply the coefficients

Multiply \(2.32\) and \(4\).
\(2.32 \times 4 = 9.28\)

Step 2: Add the exponents of 10

Add \(-6\) and \(-5\).
\(-6 + (-5) = -11\)

Step 3: Combine

The coefficient \(9.28\) is between \(1\) and \(10\), so we combine:
\(9.28 \times 10^{-11}\)

Answer:

\(1.0044 \times 10^{-3}\)

Problem 2: \((2 \times 10^{-4})(8.1 \times 10^{-1})\)