Question

Use the expansion to evaluate (1.97)^10 correct to 2 decimal places. Explain in a simple way that i can understand.

Answer 1

To evaluate \( (1.97)^{10} \) using the binomial expansion, we can express \( 1.97 \) in a form that's easy to expand. Notice that \( 1.97 \) can be rewritten as \( 2 - 0.03 \). Therefore, we can evaluate \( (2 - 0.03)^{10} \).

The binomial theorem tells us how to expand \( (a - b)^n \). The expansion is given by:

\[
(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k
\]

For our case, \( a = 2 \), \( b = 0.03 \), and \( n = 10 \). We will expand \( (2 - 0.03)^{10} \), which leads to:

\[
(2 - 0.03)^{10} = \sum_{k=0}^{10} \binom{10}{k} (2)^{10-k} (-0.03)^k
\]

This means we need to calculate a few terms of the expansion and then sum them up.

### Step 1: Calculate a few terms of the expansion

1. For \( k=0 \):

\[
\binom{10}{0} (2)^{10} (-0.03)^0 = 1 \cdot 1024 \cdot 1 = 1024
\]

2. For \( k=1 \):

\[
\binom{10}{1} (2)^{9} (-0.03)^{1} = 10 \cdot 512 \cdot (-0.03) = -153.6
\]

3. For \( k=2 \):

\[
\binom{10}{2} (2)^{8} (-0.03)^{2} = 45 \cdot 256 \cdot (0.0009) = 11.664
\]

Now let's add these three terms:

\[
1024 - 153.6 + 11.664 = 1024 - 153.6 + 11.664
\]

Calculating the sum step-by-step:

- \( 1024 - 153.6 = 870.4 \)
- \( 870.4 + 11.664 = 882.064 \)

### Step 2: Consider further terms

Next, we can consider the \( k=3 \) term for better accuracy:

4. For \( k=3 \):

\[
\binom{10}{3} (2)^{7} (-0.03)^{3} = 120 \cdot 128 \cdot (-0.000027) = -0.3888
\]

Now adding \( k=3 \) term to our previous result:

\[
882.064 - 0.3888 = 881.6752
\]

### Step 3: Rounding to two decimal places

Finally, rounding \( 881.6752 \) to two decimal places gives us:

\[
\boxed{881.68}
\]

Thus, \( (1.97)^{10} \) evaluated using the binomial expansion and rounded to two decimal places is approximately \( 881.68 \).