# Exponents and Decimals: Products Less Than 0.1

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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