Identify the cube root: Identify the cube root of 729. To find the cube root of 729, we need to find a number that, when multiplied by itself three times, gives 729.Find the prime factors: Find the prime factors of 729. Prime factors of 729 are 3 × 3 × 3 × 3 × 3 × 3, because 729 is 3 raised to the power of 6 (3^6).Express 729 as a cube: Express 729 as a cube of a product of prime factors. 729 can be written as (3^3) × (3^3), which is the same as 27 × 27.Simplify the cube root: Simplify the cube root of 729 using the prime factorization. root(3)(729) = root(3)((3^3) × (3^3)) Since the cube root of (3^3) is 3, we can simplify this to: root(3)(729) = 3 × 3Calculate the final answer: Calculate the final answer. 3 × 3 equals 9, so the cube root of 729 is 9.

Algebra 1

Simplify radical expressions

Full solution

\sqrt[3]{729} =

Identify the cube root: Identify the cube root of $729$ . $\newline$ To find the cube root of $729$ , we need to find a number that, when multiplied by itself three times, gives $729$ .
Find the prime factors: Find the prime factors of $729$ . $\newline$ Prime factors of $729$ are $3 \times 3 \times 3 \times 3 \times 3 \times 3$ , because $729$ is $3$ raised to the power of $6$ ( $3^6$ ).
Express $729$ as a cube: Express $729$ as a cube of a product of prime factors. $\newline$ $729$ can be written as $(3^3) \times (3^3)$ , which is the same as $27 \times 27$ .
Simplify the cube root: Simplify the cube root of $729$ using the prime factorization. $\newline$ $\sqrt[3]{729} = \sqrt[3]{(3^3) \times (3^3)}$ $\newline$ Since the cube root of $(3^3)$ is $3$ , we can simplify this to: $\newline$ $\sqrt[3]{729} = 3 \times 3$
Calculate the final answer: Calculate the final answer. $\newline$ $3 \times 3$ equals $9$ , so the cube root of $729$ is $9$ .