d) $$ \frac{2 \cdot 10^{7} \cdot 4 \cdot 10^{5}}{\left(4 \cdot 10^{-2}\right)^{2}} $$

Question

UpStudy AI · Accepted Answer

To evaluate the expression:

$$
\frac{2 \cdot 10^{7} \cdot 4 \cdot 10^{5}}{\left(4 \cdot 10^{-2}\right)^{2}}
$$

Let's break it down step by step.

### Step 1: Simplify the Numerator

Multiply the numerical coefficients and combine the powers of 10:

$$
2 \cdot 4 = 8
$$
$$
10^{7} \cdot 10^{5} = 10^{7+5} = 10^{12}
$$

So, the numerator becomes:

$$
8 \cdot 10^{12}
$$

### Step 2: Simplify the Denominator

First, square both the numerical coefficient and the power of 10 inside the parentheses:

$$
(4)^2 = 16
$$
$$
\left(10^{-2}\right)^2 = 10^{-4}
$$

So, the denominator becomes:

$$
16 \cdot 10^{-4}
$$

### Step 3: Divide the Numerator by the Denominator

$$
\frac{8 \cdot 10^{12}}{16 \cdot 10^{-4}} = \frac{8}{16} \cdot \frac{10^{12}}{10^{-4}} = 0.5 \cdot 10^{12 - (-4)} = 0.5 \cdot 10^{16}
$$

### Step 4: Express in Scientific Notation

$$
0.5 \cdot 10^{16} = 5 \cdot 10^{15}
$$

### Final Answer

$$
5 \times 10^{15}
$$
The final answer is $$ 5 \times 10^{15} $$.

d) \( \frac{2 \cdot 10^{7} \cdot 4 \cdot 10^{5}}{\left(4 \cdot 10^{-2}\right)^{2}} \)