How To Solve Exponents?

1. Simplifying fractional exponents:

The base b raised to the power of n/m is equal to:

b^n/m = (^m√b)ⁿ = ^m√(bⁿ)

Example:

The base 2 raised to the power of 3/2 is equal to 1 divided by the base 2 raised to the power of 3:

2^3/2 = ²√(2³) = 2.828

2. Simplifying fractions with exponents:

Fractions with exponents:

(a / b)ⁿ = aⁿ / bⁿ

Example:

(4/3)³ = 4³ / 3³ = 64 / 27 = 2.37

3. Negative fractional exponents:

The base b raised to the power of minus n/m is equal to 1 divided by the base b raised to the power of n/m:

b^-n/m = 1 / b^n/m = 1 / (^m√b)ⁿ

Example:

The base 2 raised to the power of minus 1/2 is equal to 1 divided by the base 2 raised to the power of 1/2:

2^-1/2 = 1/2^1/2 = 1/√2 = 0.7071

4. Fractions with negative exponents:

The base a/b raised to the power of minus n is equal to 1 divided by the base a/b raised to the power of n:

(a/b)^-n = 1 / (a/b)ⁿ = 1 / (aⁿ/bⁿ) = bⁿ/aⁿ

Example:

The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:

(2/3)^-2 = 1 / (2/3)² = 1 / (2²/3²) = 3²/2² = 9/4 = 2.25

5. 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 ⋅ 3^4/3 = √(2³) ⋅ ³√(3⁴) =2.828 ⋅ 4.327 = 12.237

6. 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 = 9.481

7. Dividing fractional exponents:

Dividing fractional exponents with same fractional exponent:

a ^n/m / b ^n/m = (a / b) ^n/m

Example:

3^3/2 / 2^3/2 = (3/2)^3/2 = 1.5^3/2 =√(1.5³) = √3.375 = 1.837

Dividing 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) = 2^(1/6) =⁶√2 = 1.122

Dividing fractional exponents with different exponents and fractions:

a ^n/m / b ^k/j

Example:

2^3/2 / 3^4/3 = √(2³) / ³√(3⁴) =2.828 / 4.327 = 0.654

8. Dividing fractions with exponents:

Dividing 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)¹ = 4/3 = 1.333

Dividing fractions with exponents with same exponent:

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

Example:

(4/3)³ / (3/5)³ = ((4/3)/(3/5))³= ((4⋅5)/(3⋅3))³ = (20/9)³ = 10.97

Dividing 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 = 9.481

9. Adding fractional exponents:

Adding fractional exponents is done by raising each exponent first and then adding:

a^n/m + b^k/j

Example:

3^3/2 + 2^5/2 = √(3³) + √(2⁵) = √(27) + √(32) = 5.196 + 5.657 = 10.853

Adding same bases b and exponents n/m:

b^n/m + b^n/m = 2b^n/m

Example:

4^2/3 + 4^2/3 = 2⋅4^2/3 = 2 ⋅³√(4²) = 5.04

10. Subtracting fractional exponents:

Subtracting fractional exponents is done by raising each exponent first and then subtracting:

a^n/m - b^k/j

Example:

3^3/2 - 2^5/2 = √(3³) - √(2⁵) = √(27) - √(32) = 5.196 - 5.657 = -0.488

Subtracting same bases b and exponents n/m:

3b^n/m - b^n/m = 2b^n/m

Example:

3⋅4^2/3 - 4^2/3 = 2⋅4^2/3 = 2 ⋅³√(4²) = 5.04