Identify Number and Value: Identify the number we are taking the cube root of and the value of the cube root. In root(3)(-128), we are looking for a number which, when multiplied by itself three times, gives -128.Express -128 as Term: Express -128 as a term raised to the power of 3. -128 can be written as -1 * 128, and 128 is 2 multiplied by itself 7 times (2^7). So, -128 = -1 * (2^7).Find Cube Root: Since we are looking for the cube root, we need to find a number that when raised to the power of 3 gives us -1 * (2^7). The cube root of -1 is -1, because (-1)^3 = -1. The cube root of 2^7 is 2^(7/3) because (2^(7/3))^3 = 2^(7/3 * 3) = 2^7.Combine Cube Roots: Combine the cube roots we found in the previous step. The cube root of -1 is -1, and the cube root of 2^7 is 2^(7/3). Therefore, the cube root of -128 is -1 * 2^(7/3).Simplify Expression: Simplify the expression -1 * 2^(7/3). Since 7/3 is the same as 2 + 1/3, we can express 2^(7/3) as 2^(2 + 1/3). This is the same as 2^2 * 2^(1/3), which is 4 * cube root of 2.Calculate Final Value: Calculate the final value. The cube root of -128 is -1 * 4 * cube root of 2. Since the cube root of 2 is just 2^(1/3), we can write the final answer as -4 * 2^(1/3).

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$\sqrt[3]{-128}$

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Algebra 2

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\sqrt[3]{-128}

Identify Number and Value: Identify the number we are taking the cube root of and the value of the cube root. $\newline$ In $\sqrt[3]{-128}$ , we are looking for a number which, when multiplied by itself three times, gives $-128$ .
Express $-128$ as Term: Express $-128$ as a term raised to the power of $3$ . $\newline$ $-128$ can be written as $-1 \times 128$ , and $128$ is $2$ multiplied by itself $7$ times $(2^7)$ . $\newline$ So, $-128 = -1 \times (2^7)$ .
Find Cube Root: Since we are looking for the cube root, we need to find a number that when raised to the power of $3$ gives us $-1 \times (2^7)$ . $\newline$ The cube root of $-1$ is $-1$ , because $(-1)^3 = -1$ . $\newline$ The cube root of $2^7$ is $2^{(7/3)}$ because $(2^{(7/3)})^3 = 2^{(7/3 \times 3)} = 2^7$ .
Combine Cube Roots: Combine the cube roots we found in the previous step. $\newline$ The cube root of $-1$ is $-1$ , and the cube root of $2^7$ is $2^{7/3}$ . $\newline$ Therefore, the cube root of $-128$ is $-1 \times 2^{7/3}$ .
Simplify Expression: Simplify the expression $-1 \times 2^{\frac{7}{3}}$ . Since $\frac{7}{3}$ is the same as $2 + \frac{1}{3}$ , we can express $2^{\frac{7}{3}}$ as $2^{2 + \frac{1}{3}}$ . This is the same as $2^2 \times 2^{\frac{1}{3}}$ , which is $4 \times$ cube root of $2$ .
Calculate Final Value: Calculate the final value. $\newline$ The cube root of $-128$ is $-1 \times 4 \times \text{cube root of } 2$ . $\newline$ Since the cube root of $2$ is just $2^{1/3}$ , we can write the final answer as $-4 \times 2^{1/3}$ .