Powers with Decimal Numbers: Complete Guide

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.

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Table of Contents

• 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

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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...:

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

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Raising decimals to powers

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

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) ✓

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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:

• $0.1^3=0.1\cdot 0.1\cdot 0.1=0.001$
• $0.2^3=0.2\cdot 0.2\cdot 0.2=0.008$
• $0.5^3=0.5\cdot 0.5\cdot 0.5=0.125$
• $2^3=2\cdot 2\cdot 2=8$

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Rule for moving the decimal point

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

Examples

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Fractions and decimal powers

Decimal numbers are often fractions in disguise:

Therefore:

Why is this useful?

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

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Multiplying decimal powers

With the same base

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

With different bases

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

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Common mistakes

Mistake 1: Wrong number of decimal places

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 $0.5$

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Practical applications

Area calculations

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

Probability

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

Physics

The intensity of light passing through two polarizing filters:

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Frequently Asked Questions

Why is $0.5^2=0.25$?

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

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.

$0.3^2=0.09$

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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: $0.125$

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Practice

Test yourself with interactive exercises:

• 🔢 Basic powers
• 🧮 Calculating with powers
• 📊 Powers decimal places