Question

multiply and express the answer scientific notation.
1 ((4.3 \times 10^{-6}) cdot (1.02 \times 10^{5}))
2 ((5.2 \times 10^{7}) cdot (6.1 \times 10^{1}))
3 ((3.7 \times 10^{-3}) cdot (4.23 \times 10^{3}))
4 ((6 \times 10^{-15}) cdot (1.5 \times 10^{-8}))

Explanation:

Problem 1: \((4.3 \times 10^{-6}) \cdot (1.02 \times 10^{5})\)

Step 1: Multiply the coefficients

Multiply \(4.3\) and \(1.02\).
\(4.3 \times 1.02 = 4.386\)

Step 2: Multiply the powers of 10

Use the rule \(10^{a} \times 10^{b}=10^{a + b}\). Here, \(a=-6\) and \(b = 5\), so \(10^{-6}\times10^{5}=10^{-6 + 5}=10^{-1}\)

Step 3: Combine the results

Combine the coefficient and the power of 10: \(4.386\times10^{-1}\)

Step 1: Multiply the coefficients

Multiply \(5.2\) and \(6.1\).
\(5.2\times6.1 = 31.72\)

Step 2: Multiply the powers of 10

Using \(10^{a}\times10^{b}=10^{a + b}\), with \(a = 7\) and \(b=1\), we get \(10^{7}\times10^{1}=10^{7 + 1}=10^{8}\)

Step 3: Adjust to scientific notation

Since \(31.72>10\), we rewrite \(31.72\) as \(3.172\times10^{1}\). Then multiply by \(10^{8}\): \(3.172\times10^{1}\times10^{8}=3.172\times10^{9}\)

Step 1: Multiply the coefficients

Multiply \(3.7\) and \(4.23\).
\(3.7\times4.23=15.651\)

Step 2: Multiply the powers of 10

Using \(10^{a}\times10^{b}=10^{a + b}\), with \(a=- 3\) and \(b = 3\), we get \(10^{-3}\times10^{3}=10^{-3 + 3}=10^{0}=1\)

Step 3: Combine the results

Multiply the coefficient by the power of 10: \(15.651\times1 = 15.651\). Since \(15.651>10\), rewrite it as \(1.5651\times10^{1}\)

Answer:

\(4.386\times10^{-1}\)

Problem 2: \((5.2 \times 10^{7}) \cdot (6.1 \times 10^{1})\)