Simplify the following expression to Scientific Notation: (4×10^−9)^5

Question

(2 points)
Responses

20×10−4
20 times 10 to the negative 4 power

1.024×10−48
1 point 0 2 4 times 10 to the negative 48 power

4×10−45
4 times 10 to the negative 45 power

2.0×10−3
2 point 0 times 10 to the negative 3 power

2.0×10−46
2 point 0 times 10 to the negative 46 power

4×10−20
4 times 10 to the negative 20 power

1.024×10−42
1 point 0 2 4 times 10 to the negative 42 power
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Answer 1

To simplify the expression \((4 \times 10^{-9})^5\), we will apply the property of exponents \((a \times b)^n = a^n \times b^n\).

Here, we break it down as follows:

\[ (4 \times 10^{-9})^5 = 4^5 \times (10^{-9})^5 \]

Calculating \(4^5\):

\[ 4^5 = 1024 \]

Calculating \((10^{-9})^5\):

\[ (10^{-9})^5 = 10^{-45} \]

Now, we can combine these results:

\[ (4 \times 10^{-9})^5 = 1024 \times 10^{-45} \]

Next, we need to express \(1024\) in scientific notation. \(1024\) can be rewritten as:

\[ 1024 = 1.024 \times 10^3 \]

Now, substituting back into our expression gives:

\[ 1.024 \times 10^3 \times 10^{-45} \]

When multiplying the powers of ten, we combine the exponents:

\[ 1.024 \times 10^{3 - 45} = 1.024 \times 10^{-42} \]

Thus, the simplified form in scientific notation is:

\[ \boxed{1.024 \times 10^{-42}} \]