Find the cubic root of 96.(1 point) Responses 212−−√ 2 start root 12 end root 212−−√3 2 root index 3 start root 12 end root 46–√ 4 start root 6 end root 27–√3
The cubic root of 96 is 4.
To find the cubic root of 96, follow these steps:
Step 1: Prime factorize the number 96:
96 = 2 * 2 * 2 * 2 * 2 * 3
Step 2: Group the factors into sets of three:
96 = (2 * 2 * 2) * (2 * 2 * 3)
= (2^3) * (2^2 * 3)
Step 3: Simplify each group by taking one factor out of the cube:
96 = 2^3 * (2^2 * 3)
= 2^3 * 2^2 * 3^1
Step 4: Combine the common factors by adding the exponents:
96 = 2^(3+2) * 3^1
= 2^5 * 3
Step 5: Take the cubic root of each factor:
cubic root(96) = cubic root(2^5 * 3)
= cubic root(2^3 * 2^2 * 3^1)
= 2^((3/3) * 5) * 3^(1/3)
= 2^5/3 * 3^1/3
So, the cubic root of 96 is 2^(5/3) * 3^(1/3).
To find the cube root of 96, you can use a calculator or perform the calculation manually. I will explain both methods:
1. Using a calculator:
- Enter the number 96.
- Press the button for finding the cube root (∛).
2. Manual calculation:
- Start by choosing a number and cubing it until you get a value close to 96.
- Let's start with 4, which is a good initial guess.
- 4^3 = 64, which is less than 96.
- Now, let's try 5.
- 5^3 = 125, which is greater than 96.
- To refine our estimate, we can try numbers between 4 and 5.
- Let's try 4.5.
- 4.5^3 = 91.125, which is less than 96.
- Now, let's try 4.6.
- 4.6^3 = 97.336, which is very close to 96.
- We can continue this process with more decimal places for a more accurate result, or we can stop here and use 4.6 as an approximation.
- Therefore, the cube root of 96 is approximately 4.6.
So, the correct option among the given responses is "4 start root 6 end root."