Questions: Simplify. (2 × 10^(-5))^4 (2 × 10^(-5))^4= (Use scientific notation. Use the multiplication symbol in the math palette as needed.)

Transcript text: Simplify. \[ \begin{array}{l} \left(2 \times 10^{-5}\right)^{4} \\ \left(2 \times 10^{-5}\right)^{4}= \end{array} \] $\square$ (Use scientific notation. Use the multiplication syribol in the math palette as needed.)

Solution

Solution Steps

To simplify $(2 \times 10^{-5})^4$, we need to apply the power of a product rule, which states that $(a \times b)^n = a^n \times b^n$. This means we will raise both the coefficient and the power of ten to the fourth power separately.

Step 1: Apply the Power of a Product Rule

We start with the expression $(2 \times 10^{-5})^4$. According to the power of a product rule, we can separate the coefficient and the power of ten: \[ (2 \times 10^{-5})^4 = 2^4 \times (10^{-5})^4 \]

Step 2: Calculate Each Component

Next, we calculate each part: \[ 2^4 = 16 \] \[ (10^{-5})^4 = 10^{-20} \]

Step 3: Combine the Results

Now we combine the results from the previous calculations: \[ (2 \times 10^{-5})^4 = 16 \times 10^{-20} \]

Step 4: Express in Scientific Notation

The result can be expressed in scientific notation as: \[ 16 \times 10^{-20} = 1.6 \times 10^{-19} \]