Multiplying exponents

How to multiply exponents.

• Multiplying exponents with same base
• Multiplying exponents with different bases
• Multiplying negative exponents
• Multiplying fractions with exponents
• Multiplying fractional exponents
• Multiplying variables with exponents
• Multiplying square roots with exponents

Multiplying exponents with same base

For exponents with the same base, we should add the exponents:

a ⁿ ⋅ a ^m = a ^n+m

Example:

2³ ⋅ 2⁴ = 2³⁺⁴ = 2⁷ = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128

Multiplying exponents with different bases

When the bases are different and the exponents of a and b are the same, we can multiply a and b first:

a ⁿ ⋅ b ⁿ = (a ⋅ b) ⁿ

Example:

3² ⋅ 4² = (3⋅4)² = 12² = 12⋅12 = 144

When the bases and the exponents are different we have to calculate each exponent and then multiply:

a ⁿ ⋅ b ^m

Example:

3² ⋅ 4³ = 9 ⋅ 64 = 576

Multiplying negative exponents

For exponents with the same base, we can add the exponents:

a ^-n ⋅ a ^-m = a ^-(n+m) = 1 / a ^n+m

Example:

2^-3 ⋅ 2^-4 = 2^-(3+4) = 2^-7 = 1 / 2⁷ = 1 / (2⋅2⋅2⋅2⋅2⋅2⋅2) = 1 / 128 = 0.0078125

When the bases are different and the exponents of a and b are the same, we can multiply a and b first:

a ^-n ⋅ b ^-n = (a ⋅ b) ^-n

Example:

3^-2 ⋅ 4^-2 = (3⋅4)^-2 = 12^-2 = 1 / 12² = 1 / (12⋅12) = 1 / 144 = 0.0069444

When the bases and the exponents are different we have to calculate each exponent and then multiply:

a ^-n ⋅ b ^-m

Example:

3^-2 ⋅ 4^-3 = (1/9) ⋅ (1/64) = 1 / 576 = 0.0017361

Multiplying fractions with exponents

Multiplying fractions with exponents with same fraction base:

(a / b) ⁿ ⋅ (a / b) ^m = (a / b) ^n+m

Example:

(4/3)³ ⋅ (4/3)² = (4/3)³⁺² = (4/3)⁵ = 4⁵ / 3⁵ = 4.214

Multiplying fractions with exponents with same exponent:

(a / b) ⁿ ⋅ (c / d) ⁿ = ((a / b)⋅(c / d)) ⁿ

Example:

(4/3)³ ⋅ (3/5)³ = ((4/3)⋅(3/5))³ = (4/5)³ = 0.8³ = 0.8⋅0.8⋅0.8 = 0.512

Multiplying fractions with exponents with different bases and exponents:

(a / b) ⁿ ⋅ (c / d) ^m

Example:

(4/3)³ ⋅ (1/2)² = 2.37 ⋅ 0.25 = 0.5925

Multiplying fractional exponents

Multiplying fractional exponents with same fractional exponent:

a ^n/m ⋅ b ^n/m = (a ⋅ b) ^n/m

Example:

2^3/2 ⋅ 3^3/2 = (2⋅3)^3/2 = 6^3/2 = √(6³) = √216 = 14.7

Multiplying fractional exponents with same base:

a ^(n/m) ⋅ a ^(k/j) = a ^{[(n/m)+(k/j)]}

Example:

2^(3/2) ⋅ 2^(4/3) = 2^{[(3/2)+(4/3)]} = 7.127

Multiplying fractional exponents with different exponents and fractions:

a ^n/m ⋅ b ^k/j

Example:

2 ^3/2 ⋅ 2^4/3 = √(2³) ⋅ ³√(2⁴) = 2.828 ⋅ 2.52 = 7.127

Multiplying square roots with exponents

For exponents with the same base, we can add the exponents:

(√a)ⁿ ⋅ (√a)^m = a^(n+m)/2

Example:

(√5)² ⋅ (√5)⁴ = 5^(2+4)/2 = 5^6/2 = 5³ = 125

Multiplying variables with exponents

For exponents with the same base, we can add the exponents:

xⁿ ⋅ x^m = x^n+m

Example:

x² ⋅ x³ = (x⋅x) ⋅ (x⋅x⋅x) = x²⁺³ = x⁵