Powers and scientific notation

For a shared base greater than 1, the greater exponent gives the greater value. To find the factor between them, divide the powers and subtract exponents: aⁿ ÷ aᵐ = aⁿ⁻ᵐ. A multiplicative comparison uses a ratio, not a difference.

A negative exponent does not mean that the number itself is negative. When a ≠ 0, use the rule a⁻ⁿ = 1 / aⁿ.

1. Keep the same base a.
2. Move the power to the denominator and make the exponent positive n.
3. Check that the denominator is not zero.

Thus a negative exponent takes the reciprocal of a power; it does not place a minus sign in front of its value.

Scientific notation is a × 10ⁿ with 1 ≤ a < 10. Move the decimal point so that one nonzero digit remains before it; the number of places moved determines the exponent. For example, 50 000 = 5 × 10⁴.

For a shared base greater than 1, the greater exponent gives the greater value. To find the factor between them, divide the powers and subtract exponents: aⁿ ÷ aᵐ = aⁿ⁻ᵐ. A multiplicative comparison uses a ratio, not a difference.

The power aⁿ is the product of n factors equal to a. For example, 3⁴ = 3 × 3 × 3 × 3 = 81. Do not simply multiply the base by the exponent.

When multiplying powers with the same base, keep the base and add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 2³ × 2⁴ = 2⁷.

Evaluate the power first, then add the other number. For example, 2³ + 5 = 8 + 5 = 13. The exponent applies only to its base.

When dividing powers with the same nonzero base, subtract the second exponent from the first: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. For example, 3⁵ ÷ 3² = 3³.

Scientific notation is a × 10ⁿ with 1 ≤ a < 10. Move the decimal point so that one nonzero digit remains before it; the number of places moved determines the exponent. For example, 50 000 = 5 × 10⁴.

When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ. For example, (2³)² = 2⁶.