To find the cube root of 2146689, you can use a method called prime factorization.
Step 1: Prime factorize 2146689
To do this, divide the number by the smallest prime number possible, which is 3.
2146689 ÷ 3 = 715563
Repeat the process with the new number until you cannot divide by 3 anymore:
715563 ÷ 3 = 238521
238521 ÷ 3 = 79507
79507 ÷ 3 = 26502.333
We stop here since we need a whole number for the next step.
We can see that the prime factorization of 2146689 is:
2146689 = 3^6 × 7^2 × 11^2
Step 2: Determine the cube root of each prime factor
The cube root of 3^6 is 3^2 = 9, since 3^2 × 3^2 × 3^2 = (3 × 3 × 3) × (3 × 3 × 3) × (3 × 3 × 3) = 3^6.
The cube root of 7^2 is 7^(2/3), which can be found using a calculator or by estimating. A good estimate is to round 7^(2/3) to the nearest whole number. In this case, 7^(2/3) is close to 3, so we will use that as an estimate.
The cube root of 11^2 is 11^(2/3), which can also be estimated. 11^(2/3) is close to 2.8, so we will use that as an estimate.
Step 3: Multiply the cube roots of the prime factors
To find the cube root of the original number, we multiply the cube roots of the prime factors:
Cube root of 2146689 = Cube root of (3^6 × 7^2 × 11^2)
= Cube root of 3^6 × Cube root of 7^2 × Cube root of 11^2
= 9 × 3 × 2.8
= 75.6
Therefore, the cube root of 2146689 is approximately 75.6.
To find the cube root of 2,146,689 step by step, you can use an approximation method such as the Newton-Raphson method. Here's how you can do it:
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.Step 2: Use the following formula to improve the guess:
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.Step 2: Use the following formula to improve the guess:new_guess = (2 * old_guess + number / (old_guess * old_guess)) / 3
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.Step 2: Use the following formula to improve the guess:new_guess = (2 * old_guess + number / (old_guess * old_guess)) / 3Step 3: Repeat step 2 several times, using the new_guess as the old_guess each time, until the guess converges to a value close enough to the cube root.
Let's go through the steps:
1. Start with an initial guess: old_guess = 1.
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guess
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guessnew_guess = (2 * 1 + 2,146,689 / (1 * 1)) / 3
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guessnew_guess = (2 * 1 + 2,146,689 / (1 * 1)) / 3 = (2 + 2,146,689) / 3
Answers & Comments
Answer:
75.6
Step-by-step explanation:
To find the cube root of 2146689, you can use a method called prime factorization.
Step 1: Prime factorize 2146689
To do this, divide the number by the smallest prime number possible, which is 3.
2146689 ÷ 3 = 715563
Repeat the process with the new number until you cannot divide by 3 anymore:
715563 ÷ 3 = 238521
238521 ÷ 3 = 79507
79507 ÷ 3 = 26502.333
We stop here since we need a whole number for the next step.
We can see that the prime factorization of 2146689 is:
2146689 = 3^6 × 7^2 × 11^2
Step 2: Determine the cube root of each prime factor
The cube root of 3^6 is 3^2 = 9, since 3^2 × 3^2 × 3^2 = (3 × 3 × 3) × (3 × 3 × 3) × (3 × 3 × 3) = 3^6.
The cube root of 7^2 is 7^(2/3), which can be found using a calculator or by estimating. A good estimate is to round 7^(2/3) to the nearest whole number. In this case, 7^(2/3) is close to 3, so we will use that as an estimate.
The cube root of 11^2 is 11^(2/3), which can also be estimated. 11^(2/3) is close to 2.8, so we will use that as an estimate.
Step 3: Multiply the cube roots of the prime factors
To find the cube root of the original number, we multiply the cube roots of the prime factors:
Cube root of 2146689 = Cube root of (3^6 × 7^2 × 11^2)
= Cube root of 3^6 × Cube root of 7^2 × Cube root of 11^2
= 9 × 3 × 2.8
= 75.6
Therefore, the cube root of 2146689 is approximately 75.6.
step -ßy-step explanation:☞
To find the cube root of 2,146,689 step by step, you can use an approximation method such as the Newton-Raphson method. Here's how you can do it:
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.Step 2: Use the following formula to improve the guess:
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.Step 2: Use the following formula to improve the guess:new_guess = (2 * old_guess + number / (old_guess * old_guess)) / 3
Step 1: Start with an initial guess for the cube root. Let's say we start with a guess of 1.Step 2: Use the following formula to improve the guess:new_guess = (2 * old_guess + number / (old_guess * old_guess)) / 3Step 3: Repeat step 2 several times, using the new_guess as the old_guess each time, until the guess converges to a value close enough to the cube root.
Let's go through the steps:
1. Start with an initial guess: old_guess = 1.
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guess
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guessnew_guess = (2 * 1 + 2,146,689 / (1 * 1)) / 3
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guessnew_guess = (2 * 1 + 2,146,689 / (1 * 1)) / 3 = (2 + 2,146,689) / 3
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guessnew_guess = (2 * 1 + 2,146,689 / (1 * 1)) / 3 = (2 + 2,146,689) / 3 = 2,146,691 / 3
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guessnew_guess = (2 * 1 + 2,146,689 / (1 * 1)) / 3 = (2 + 2,146,689) / 3 = 2,146,691 / 3 ≈ 715,563.6667
1. Start with an initial guess: old_guess = 1.2. Calculate the new_guessnew_guess = (2 * 1 + 2,146,689 / (1 * 1)) / 3 = (2 + 2,146,689) / 3 = 2,146,691 / 3 ≈ 715,563.66673. Update the old_guess to be the new_guess: old_guess = 715,563.6667
e new_guess: old_guess = 715,563.66674. Repeat step 2 with the new_guess as the old_guess:
e new_guess: old_guess = 715,563.66674. Repeat step 2 with the new_guess as the old_guess:new_guess = (2 * 715,563.6667 + 2,146,689 / (715,563.6667 * 715,563.6667)) / 3
e new_guess: old_guess = 715,563.66674. Repeat step 2 with the new_guess as the old_guess:new_guess = (2 * 715,563.6667 + 2,146,689 / (715,563.6667 * 715,563.6667)) / 3 ≈ 715,562.6741
e new_guess: old_guess = 715,563.66674. Repeat step 2 with the new_guess as the old_guess:new_guess = (2 * 715,563.6667 + 2,146,689 / (715,563.6667 * 715,563.6667)) / 3 ≈ 715,562.67415. Update the old_guess: old_guess = 715,562.6741.
e new_guess: old_guess = 715,563.66674. Repeat step 2 with the new_guess as the old_guess:new_guess = (2 * 715,563.6667 + 2,146,689 / (715,563.6667 * 715,563.6667)) / 3 ≈ 715,562.67415. Update the old_guess: old_guess = 715,562.6741.6. Repeat step 2 again:
e new_guess: old_guess = 715,563.66674. Repeat step 2 with the new_guess as the old_guess:new_guess = (2 * 715,563.6667 + 2,146,689 / (715,563.6667 * 715,563.6667)) / 3 ≈ 715,562.67415. Update the old_guess: old_guess = 715,562.6741.6. Repeat step 2 again:new_guess = (2 * 715,562.6741 + 2,146,689 / (715,562.6741 * 715,562.6741)) / 3
e new_guess: old_guess = 715,563.66674. Repeat step 2 with the new_guess as the old_guess:new_guess = (2 * 715,563.6667 + 2,146,689 / (715,563.6667 * 715,563.6667)) / 3 ≈ 715,562.67415. Update the old_guess: old_guess = 715,562.6741.6. Repeat step 2 again:new_guess = (2 * 715,562.6741 + 2,146,689 / (715,562.6741 * 715,562.6741)) / 3 ≈ 715,562.6741
At this point, the new_guess is approximately the same as the old_guess. Therefore, the cube root of 2,146,689 is approximately 715,562.6741.
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