Powers with Decimal Numbers: How to Raise Decimals

Decimals are raised to powers just like whole numbers – you just need to be careful with the decimal point. Let's explain how to do it.

What is a decimal number
Raising decimals to powers
Examples of basic powers
Rule for moving the decimal point
Fractions and decimal powers
Multiplying decimal powers
Common mistakes
Practical applications
Frequently Asked Questions

What is a decimal number

A decimal number is a number that contains a decimal point, which separates the whole part from the fractional part.

Examples: $0.5$ ; $0.25$ ; $1.3$ ; $3.14$

Decimal numbers are actually fractions with denominators of 10, 100, 1000...:

$0.5 = \frac{5}{10} = \frac{1}{2}$
$0.25 = \frac{25}{100} = \frac{1}{4}$
$0.1 = \frac{1}{10}$

Raising decimals to powers

Raising decimals to powers works the same way as with whole numbers:

(0.5)^{2} = 0.5 \cdot 0.5

But here it's important to know where to place the decimal point in the result.

Basic rule

Number of decimal places in the result = number of decimal places in the base × exponent

Example:

(0.5)^{2}

0.5 has 1 decimal place
exponent is 2
result will have 1 × 2 = 2 decimal places
$0.5 \cdot 0.5 = 0.25$ (2 decimal places) ✓

Examples of basic powers

Square (second power)

Squaring decimal numbers:

$0. 1^{2} = 0.1 \cdot 0.1 = 0.01$
$0. 2^{2} = 0.2 \cdot 0.2 = 0.04$
$0. 5^{2} = 0.5 \cdot 0.5 = 0.25$
$0. 3^{2} = 0.3 \cdot 0.3 = 0.09$
$1. 5^{2} = 1.5 \cdot 1.5 = 2.25$

Cube (third power)

Cubing decimal numbers:

$0. 1^{3} = 0.1 \cdot 0.1 \cdot 0.1 = 0.001$
$0. 2^{3} = 0.2 \cdot 0.2 \cdot 0.2 = 0.008$
$0. 5^{3} = 0.5 \cdot 0.5 \cdot 0.5 = 0.125$
$2^{3} = 2 \cdot 2 \cdot 2 = 8$

Rule for moving the decimal point

For powers of the number 10, there's a simple rule:

(0.1)^{n} = 1 0^{- n}

(0.01)^{n} = 1 0^{- 2 n}

Examples

0. 1^{2} = 0.01 = 1 0^{- 2}

0. 1^{3} = 0.001 = 1 0^{- 3}

0.0 1^{2} = 0.0001 = 1 0^{- 4}

Fractions and decimal powers

Decimal numbers are often fractions in disguise:

0.5 = \frac{1}{2}

Therefore:

(0.5)^{2} = (\frac{1}{2})^{2} = \frac{1 ^{2}}{2 ^{2}} = \frac{1}{4} = 0.25

(0.25)^{2} = (\frac{1}{4})^{2} = \frac{1}{16} = 0.0625

Why is this useful?

Knowing that $0.5 = \frac{1}{2}$ helps us understand why $0. 5^{2} = 0.25$ :

(\frac{1}{2})^{2} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} = 0.25

Multiplying decimal powers

With the same base

When multiplying powers with the same base, we add the exponents:

(0.5)^{2} \cdot (0.5)^{3} = (0.5)^{2 + 3} = (0.5)^{5} = 0.03125

With different bases

When multiplying powers with different bases, we multiply the bases:

(0.2)^{2} \cdot (0.5)^{2} = (0.2 \cdot 0.5)^{2} = (0.1)^{2} = 0.01

Common mistakes

Mistake 1: Wrong number of decimal places

(0.3)^{2} = 0.09 not 0.9

One decimal place × 2 = two decimal places

Mistake 2: Confusing $0. 1^{2}$ with $0.1 \cdot 2$

0. 1^{2} = 0.1 \cdot 0.1 = 0.01

0.1 \cdot 2 = 0.2

(this is multiplication by 2, not squaring!)

Mistake 3: Forgetting that decimals are fractions

0. 5^{2} = 0.25

, not

0.5

Practical applications

Area calculations

If a square has a side length of $0.5$ m, its area is:

A = (0.5)^{2} = 0.25 m^{2}

Probability

The probability of getting heads twice in a row when flipping a fair coin:

P = (0.5)^{2} = 0.25 = 25%

Physics

The intensity of light passing through two polarizing filters:

I = I_{0} \cdot (0.5)^{2} = 0.25 \cdot I_{0}

Frequently Asked Questions

Why is $0. 5^{2} = 0.25$ ?

Because $0.5 = \frac{1}{2}$ , and $(\frac{1}{2})^{2} = \frac{1}{4} = 0.25$ .

Is $0. 1^{2} = 0.1 \cdot 2$ ?

No! $0. 1^{2} = 0.1 \cdot 0.1 = 0.01$

0.1 \cdot 2 = 0.2

These are completely different operations.

How many decimal places should $0. 3^{2}$ have?

0.3

has 1 decimal place, exponent is 2, so result has 1 × 2 = 2 decimal places.

0. 3^{2} = 0.09

Summary

Expression: $0. 1^{2}$ → Calculation: $0.1 \cdot 0.1$ → Result: $0.01$
Expression: $0. 2^{2}$ → Calculation: $0.2 \cdot 0.2$ → Result: $0.04$
Expression: $0. 5^{2}$ → Calculation: $0.5 \cdot 0.5$ → Result: $0.25$
Expression: $0. 1^{3}$ → Calculation: $0.1 \cdot 0.1 \cdot 0.1$ → Result: $0.001$
Expression: $0. 2^{3}$ → Calculation: $0.2 \cdot 0.2 \cdot 0.2$ → Result: $0.008$
Expression: $0. 5^{3}$ → Calculation: $0.5 \cdot 0.5 \cdot 0.5$ → Result: $0.125$

Practice

Test yourself with interactive exercises: