Question

write the number in decimal notation.
2.4 × 10⁻⁹
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1. - / 2 points

evaluate the expression without using a calculator. (enter your answer in scientific notation.)
(3 × 10⁶)³
× 10

Explanation:

Problem 1: Write \( 2.4 \times 10^{-9} \) in decimal notation

Step-by-Step Explanation:

To convert a number from scientific notation \( a \times 10^n \) to decimal notation:

• If \( n \) is negative, move the decimal point in \( a \) to the left by \( |n| \) places.

For \( 2.4 \times 10^{-9} \):

• \( 10^{-9} \) means we move the decimal in \( 2.4 \) left 9 places.
• Start with \( 2.4 \), add 8 leading zeros (since we need 9 total moves: 1 move to get to \( 0.24 \), then 8 more zeros).

Final Answer (Problem 1):

\( 0.0000000024 \)

Problem 2: Evaluate \( (3 \times 10^6)^3 \) in scientific notation

Step-by-Step Explanation:

Use the power - of - a - product rule: \( (ab)^n=a^n\times b^n \), and the power - of - a - power rule: \( (a^m)^n=a^{m\times n} \).

1. Apply the power - of - a - product rule to \( (3\times10^{6})^{3} \):

We get \( 3^{3}\times(10^{6})^{3} \).

1. Calculate \( 3^{3} \):

\( 3^{3}=3\times3\times3 = 27 \).

1. Apply the power - of - a - power rule to \( (10^{6})^{3} \):

Using \( (a^{m})^{n}=a^{m\times n} \), we have \( (10^{6})^{3}=10^{6\times3}=10^{18} \).

1. Now we have \( 27\times10^{18} \). But in scientific notation, the coefficient \( a \) must satisfy \( 1\leq a<10 \). So we rewrite \( 27 \) as \( 2.7\times10^{1} \).

1. Then \( 27\times10^{18}=2.7\times10^{1}\times10^{18} \). Using the rule of exponents \( a^{m}\times a^{n}=a^{m + n} \), we get \( 2.7\times10^{1 + 18}=2.7\times10^{19} \).

Final Answer (Problem 2):

\( 2.7\times10^{19} \)

Answer:

Problem 1: Write \( 2.4 \times 10^{-9} \) in decimal notation

Step-by-Step Explanation:

To convert a number from scientific notation \( a \times 10^n \) to decimal notation:

• If \( n \) is negative, move the decimal point in \( a \) to the left by \( |n| \) places.

For \( 2.4 \times 10^{-9} \):

• \( 10^{-9} \) means we move the decimal in \( 2.4 \) left 9 places.
• Start with \( 2.4 \), add 8 leading zeros (since we need 9 total moves: 1 move to get to \( 0.24 \), then 8 more zeros).

Final Answer (Problem 1):

\( 0.0000000024 \)

Problem 2: Evaluate \( (3 \times 10^6)^3 \) in scientific notation

Step-by-Step Explanation:

Use the power - of - a - product rule: \( (ab)^n=a^n\times b^n \), and the power - of - a - power rule: \( (a^m)^n=a^{m\times n} \).

1. Apply the power - of - a - product rule to \( (3\times10^{6})^{3} \):

We get \( 3^{3}\times(10^{6})^{3} \).

1. Calculate \( 3^{3} \):

\( 3^{3}=3\times3\times3 = 27 \).

1. Apply the power - of - a - power rule to \( (10^{6})^{3} \):

Using \( (a^{m})^{n}=a^{m\times n} \), we have \( (10^{6})^{3}=10^{6\times3}=10^{18} \).

1. Now we have \( 27\times10^{18} \). But in scientific notation, the coefficient \( a \) must satisfy \( 1\leq a<10 \). So we rewrite \( 27 \) as \( 2.7\times10^{1} \).

1. Then \( 27\times10^{18}=2.7\times10^{1}\times10^{18} \). Using the rule of exponents \( a^{m}\times a^{n}=a^{m + n} \), we get \( 2.7\times10^{1 + 18}=2.7\times10^{19} \).

Final Answer (Problem 2):

\( 2.7\times10^{19} \)