Identify the cube root: Identify the cube root of 512. To find the cube root of 512, we need to find a number that, when multiplied by itself three times, gives 512.Find the prime factors: Find the prime factors of 512. Prime factors of 512 can be found by dividing the number by the smallest prime number until we reach 1. 512 ÷ 2 = 256 256 ÷ 2 = 128 128 ÷ 2 = 64 64 ÷ 2 = 32 32 ÷ 2 = 16 16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 = 2 2 ÷ 2 = 1 So, the prime factors of 512 are 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (nine 2's).Group the prime factors: Group the prime factors into triples. Since we are looking for the cube root, we group the prime factors into triples: (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2). Each group of three 2's is the cube of 2, so we have 2^3 × 2^3 × 2^3.Simplify the expression: Simplify the expression. The cube root of 2^3 is 2, so the cube root of 2^3 × 2^3 × 2^3 is 2 × 2 × 2. Therefore, root(3)(512) = 2 × 2 × 2 = 8.

Algebra 1

Simplify radical expressions

Full solution

\sqrt[3]{512}=

Identify the cube root: Identify the cube root of $512$ . $\newline$ To find the cube root of $512$ , we need to find a number that, when multiplied by itself three times, gives $512$ .
Find the prime factors: Find the prime factors of $512$ . $\newline$ Prime factors of $512$ can be found by dividing the number by the smallest prime number until we reach $1$ . $\newline$ $512 \div 2 = 256$ $\newline$ $256 \div 2 = 128$ $\newline$ $128 \div 2 = 64$ $\newline$ $64 \div 2 = 32$ $\newline$ $32 \div 2 = 16$ $\newline$ $16 \div 2 = 8$ $\newline$ $8 \div 2 = 4$ $\newline$ $512$ $0$ $\newline$ $512$ $1$ $\newline$ So, the prime factors of $512$ are $512$ $3$ (nine $512$ $4$ 's).
Group the prime factors: Group the prime factors into triples. $\newline$ Since we are looking for the cube root, we group the prime factors into triples: $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)$ . $\newline$ Each group of three $2$ 's is the cube of $2$ , so we have $2^3 \times 2^3 \times 2^3$ .
Simplify the expression: Simplify the expression. $\newline$ The cube root of $2^3$ is $2$ , so the cube root of $2^3 \times 2^3 \times 2^3$ is $2 \times 2 \times 2$ . $\newline$ Therefore, $\sqrt[3]{512} = 2 \times 2 \times 2 = 8$ .