# Exponents and Decimals: Products Less Than 0.1

In numbers such as (0.44)^{3}, the decimal 0.44 is the base and 3 is the **exponent**. Such numbers are repeated products of the base.

**Rules**

We see that raising a decimal base to an exponent is same as multiplying the decimal by itself as many times as the exponent.

We treat the decimals as whole numbers by ignoring the decimal points and multiply.

After counting the total number of decimal places in these numbers, we put a decimal point after that many places from the right in the answer.

Here we are considering exponents of decimals where the products are less than 0.1.

Evaluate (0.31)^{2}

### Solution

**Step 1:**

Consider (0.31)^{2}; Here, we are raising the decimal base 0.31 to a power of 2.

(0.31)^{2} = 0.31 × 0.31 = 0.0961

**Step 2:**

We see that squaring a decimal base is in fact same as multiplying the decimal by itself. We treat the decimals as whole numbers by ignoring the decimal points and multiply.

31 × 31 = 961

**Step 3:**

After counting the total number of decimal places which is four in these numbers, we put a decimal point after four places from the right in the answer.

So, 0.31 × 0.31 = 0.0961

We see that that the product is less than 0.1

Evaluate (0.46)^{3}

### Solution

**Step 1:**

Consider (0.46)^{3}; Here, we are raising the decimal base 0.46 to a power of 3.

(0.46)^{3} = 0.46 × 0.46 × 0.46

**Step 2:**

We see that cubing a decimal base is in fact same as multiplying the decimal by itself three times. We treat the decimals as whole numbers by ignoring the decimal points and multiply.

46 × 46 × 46 = 97336

**Step 3:**

After counting the total number of decimal places which is six in these numbers, we put a decimal point after six places from the right in the answer.

So, 0.46 × 0.46 × 0.46 = 0.097336

We see that that the product is less than 0.1