Properties of Roots and Exponents Calculator

Property 1. To raise the number a to the power k/n - means to extract the root of the degree n from the number a to the power k.

a ⩾ 0
n ⩾ 2, n ∈ N, N = {1, 2, 3...}
k ∈ Z, Z = {-3, -2, -1, 0, 1, 2, 3...}

Property 2. The power of a number a with a negative exponent is one divided by the power of the same number with a positive exponent.

a > 0
n ∈ R

Property 3. Extracting the root of degree n from the product of numbers a and b, can be represented as the product of the root of degree n from a, and the root of degree n from b.

a ⩾ 0
b ⩾ 0
n ⩾ 2, n ∈ N, N = {1, 2, 3...}

Property 4. Extracting a root of degree n from a root of degree k of a, can be represented as a root of degree n by k of a.

a ⩾ 0
n ⩾ 2, n ∈ N, N = {1, 2, 3...}
k ⩾ 2, k ∈ N, N = {1, 2, 3...}

Property 5. The number a to the power of n + k, can be represented as the product of the numbers: a to the power of n and a to the power of k.

a ⩾ 0, a ∈ R
n ∈ Q
k ∈ Q

Property 6. The number a to the power of n-k, can be represented as the ratio of numbers: a to the power of n to a to the power of k.

a ⩾ 0, a ∈ R
n ∈ Q
k ∈ Q

Property 7. When a number a is raised to the power of n to the power of k, he base of a remains the same, and the exponents of k and n are multiplied.

a ⩾ 0, a ∈ R
n ∈ Q
k ∈ Q

Property 8. The product of numbers a and b to the power of n, can be represented as the product of the number a to the power of n and the number b to the power of n.

a ⩾ 0, a ∈ R
b ⩾ 0, b ∈ R
n ∈ Q

Property 9. The ratio of numbers a and b to the power of n, can be represented as the ratio of the number a to the power of n to the number b to the power of n.

a ⩾ 0, a ∈ R
b > 0, b ∈ R
n ∈ Q