Exponents with decimal and fractional bases

Key notes:

Understanding Exponents

• An exponent represents repeated multiplication of a base.
• Example: 2³ = 2 × 2 × 2 = 8.

Decimal Bases with Exponents

• Decimal numbers can be used as bases in exponent expressions.
• Example: (0.2)³ = 0.2 × 0.2 × 0.2 = 0.008.
• The value gets smaller when raising a decimal (between 0 and 1) to a positive exponent.

Fractional Bases with Exponents

• Fractions can be used as bases and follow the same exponent rules.
• Example: (1/2)³ = 1/2 × 1/2 × 1/2 = 18​.
• The numerator and denominator are both raised to the exponent.
• Example: (3/5)² = 3²/5² = 9/25​.

Negative Exponents with Decimal and Fractional Bases

• A negative exponent means taking the reciprocal of the base.
• Example: (0.2)⁻² = 1 / (0.2)² = 1 / 0.04 = 25
• Example: (2/3)^-2 = (32)² = 9/4.

Exponent Rules Apply

• Product Rule: a^m × aⁿ = a^m+n
• Quotient Rule: a^m / aⁿ=a^m−n
• Power Rule: (a^m)ⁿ = a^m×n
• Zero Exponent Rule: Any nonzero base raised to the power of 0 is 1 (e.g., (0.5)⁰ = 1).

Real-World Applications

• Used in scientific notation for very small numbers (e.g., 3.2 × 10⁻⁴).
• Growth and decay problems involve fractional exponent bases (e.g., interest rates and bacteria growth).

Learn with an example

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