Exponents with decimal and fractional bases

Key Notes :

🌟 Exponents with Decimal and Fractional Bases

🔹 What is an Exponent?

An exponent shows how many times a number (called the base) is multiplied by itself.

👉 Example:

3²=3×3=9

🔹 Decimal Bases

A decimal base is a number that contains a decimal point.

When a decimal base is raised to a power, we multiply the base by itself as many times as the exponent shows.

✨ Examples:

1. (0.2)²=0.2×0.2=0.04
2. (0.5)³=0.5×0.5×0.5=0.125

👉 Tip: The result becomes smaller as you multiply decimals less than 1.

🔹 Fractional Bases

A fractional base is a number written in the form a/b​.

When raised to a power, both the numerator and the denominator are raised to that power.

✨ Examples:

1. (2/3)2=2²/3²=4/9
2. (1/4)³=1³/4³=1/64

🔹 Negative Exponents

A negative exponent means we take the reciprocal of the base and make the exponent positive.

✨ Examples:

1. (0.2)⁻¹=1/0.2=5
2. (3/4)⁻²=(4/3)²=16/9

🔹 Zero Exponent Rule

Any non-zero number raised to the power of 0 equals 1.

✨ Examples:

1. (0.5)⁰=1
2. (2/3)⁰=1

🔹 Comparing Decimal and Fractional Bases

• Both decimals and fractions less than 1 become smaller as the exponent increases.
• Example: (0.5)²=0.25 and (1/2)²=1/4​, which are the same value.

🔹 Real-Life Applications 💡

• Calculating interest rates and growth/decay in financial problems.
• Finding areas and volumes when measurements are in decimals or fractions.
• Used in scientific notation for very small numbers (like (1.2×10⁻³)²).

🧠 Quick Practice

1. (0.3)³=?
2. (2/5)²=?
3. (0.4)⁻²=?
4. (3/4)⁰=?

Learn with an example

Let’s practice!🖊️