Question

approximate the following number using a calculator. ( 128e^{0.028(8)} ) ( 128e^{0.028(8)} approx square ) (round to three decimal places as needed.)

Explanation:

Step1: Calculate the exponent part

First, calculate the value of the exponent \(0.028\times8\).
\(0.028\times8 = 0.224\)

Step2: Calculate the exponential function

Next, find the value of \(e^{0.224}\). Using a calculator, \(e^{0.224}\approx1.251\) (this is an approximation, more accurately, using a calculator for \(e^{0.224}\))

Step3: Multiply by 128

Then, multiply this result by 128: \(128\times1.251 = 160.128\)? Wait, no, let's do it more accurately. First, calculate \(e^{0.224}\) more precisely. Using a calculator, \(e^{0.224}\approx e^{0.224}\). Let's compute it:

We know that \(e^x=\sum_{n = 0}^{\infty}\frac{x^n}{n!}\), but using a calculator directly: \(e^{0.224}\approx1.251037\)

Then, \(128\times1.251037 = 128\times1.251037\)

Calculate \(128\times1.251037\):

\(128\times1.251037=(100 + 20+8)\times1.251037=100\times1.251037+20\times1.251037 + 8\times1.251037\)

\(=125.1037+25.02074+10.008296\)

\(=125.1037+25.02074 = 150.12444+10.008296=160.132736\)

Wait, maybe my initial approximation of \(e^{0.224}\) was wrong. Let's use a calculator step by step.

First, compute \(0.028\times8 = 0.224\)

Then, compute \(e^{0.224}\). Using a calculator (like a scientific calculator), \(e^{0.224}\approx1.251037\)

Then, multiply by 128: \(128\times1.251037 = 128\times1.251037\)

Let's do this multiplication:

\(1.251037\times128\)

\(1.251037\times100 = 125.1037\)

\(1.251037\times20 = 25.02074\)

\(1.251037\times8 = 10.008296\)

Adding them together: \(125.1037+25.02074 = 150.12444+10.008296 = 160.132736\)

Rounding to three decimal places, we look at the fourth decimal place. The number is \(160.132736\), the fourth decimal is 7, which is greater than 5, so we round up the third decimal place. So \(160.133\)? Wait, no, wait: \(160.132736\)

The decimal places:

First decimal: 1 (tenths)

Second: 3 (hundredths)

Third: 2 (thousandths)

Fourth: 7 (ten - thousandths)

Since the fourth decimal is 7 ≥ 5, we add 1 to the third decimal place: \(2 + 1=3\)

So the number rounded to three decimal places is \(160.133\)? Wait, but let's check with a calculator directly.

Using a calculator:

First, calculate \(0.028\times8 = 0.224\)

Then, calculate \(e^{0.224}\approx1.251037\)

Then, \(128\times1.251037 = 160.132736\approx160.133\) (rounded to three decimal places)

Wait, maybe I made a mistake in the multiplication. Let's do \(128\times1.251037\):

\(1.251037\times128\)

\(128\times1 = 128\)

\(128\times0.2 = 25.6\)

\(128\times0.05 = 6.4\)

\(128\times0.001037=128\times0.001+128\times0.000037 = 0.128+0.004736 = 0.132736\)

Now, add them: \(128+25.6 = 153.6+6.4 = 160+0.132736 = 160.132736\)

Yes, so rounding to three decimal places, it's \(160.133\)

Wait, but let's check with a calculator input: 128 e^(0.0288)

Calculating 0.028*8 = 0.224

e^0.224 ≈ 1.251037

128*1.251037 = 160.132736, which rounds to 160.133

Answer:

\(160.133\)