How to Calculate 1.10^8 in a Simple Way

Calculating 1.108 can be done efficiently using various methods. The primary approach can be done manually using the properties of exponents or through the use of a calculator or programming language. Here, we explore both methods and provide a step-by-step process for the manual approach.

1. Basic Property of Exponents

One of the simplest ways to calculate 1.108 is by using the basic property of exponents that states (a^b a times a times ldots b times). While this method is straightforward, it can be cumbersome for large exponents. Therefore, for a quick calculation, use a scientific calculator or a computer program.

2. Manual Calculation with Exponent Squaring

For those who prefer a manual approach, we can break down 1.108 using the steps below:

First, calculate (1.10^2) Then, calculate (1.10^4) by squaring (1.10^2) Finally, calculate (1.10^8) by squaring (1.10^4)

Here is the step-by-step process:

Calculate (1.10^2)

(1.10^2 1.10 times 1.10 1.21)

Calculate (1.10^4)

(1.10^4 (1.10^2)^2 1.21^2 1.4641)

Calculate (1.10^8)

(1.10^8 (1.10^4)^2 approx 1.4641^2 approx 2.1466)

Thus, the approximate value of 1.108 is 2.1466, which can be computed quickly using a calculator or computer.

3. Using the Binomial Theorem

For a more precise manual calculation, the Binomial Theorem can be employed. The Binomial Theorem states that ((a b)^n sum_{k0}^{n} binom{n}{k} a^{n-k}b^k). Applying this theorem to 1.108 can yield a more accurate result.

(1.10^8 (1 0.1)^8 1 8 cdot 0.1 frac{8 cdot 7}{2!} cdot 0.1^2 frac{8 cdot 7 cdot 6}{3!} cdot 0.1^3 frac{8 cdot 7 cdot 6 cdot 5}{4!} cdot 0.1^4 ldots)

Breaking it down, we get:

First Term 1

Second Term 8 cdot 0.1 0.8

Third Term Second Term cdot 0.1 cdot frac{7}{2} 0.8 cdot 0.1 cdot 3.5 0.28

Fourth Term Third Term cdot 0.1 cdot frac{6}{3} 0.28 cdot 0.1 cdot 2 0.056

Fifth Term Fourth Term cdot 0.1 cdot frac{5}{4} 0.056 cdot 0.1 cdot 1.25 0.007

Summing these terms gives:

1 0.8 0.28 0.056 0.007 2.143

The actual value of 1.108 is 2.14358881. As you can see, the calculation becomes more intricate, but it provides a highly precise result.

Conclusion

Calculating 1.108 can be done quickly and accurately using a combination of the exponent squaring method and the Binomial Theorem. Whether you choose the calculator path or the manual paths, both methods are useful in different scenarios.