Roots and Irrational Numbers Unit Test
Math 8 Q2 (Pre-Algebra) / Roots and Irrational Numbers
rewrite the irrational cube root ∛189 as a product of an integer and another irrational cube root. show your work
To rewrite the irrational cube root ∛189 as a product of an integer and another irrational cube root, we need to find a perfect cube factor of 189.
The prime factorization of 189 is:
189 = 3 × 3 × 3 × 7
Let's break it down into perfect cubes:
189 = (3 × 3 × 3) × 7 = 27 × 7
Now, rewrite the irrational cube root ∛189:
∛189 = ∛(27 × 7)
Since 27 is a perfect cube, we can rewrite it as:
∛189 = ∛(27 × 7) = ∛27 × ∛7
The cube root of 27 is 3, so we have:
∛27 = 3
Therefore, we can rewrite the irrational cube root ∛189 as a product of an integer and another irrational cube root:
∛189 = 3∛7
put it in first person pov when your talking about solving it and put it in complete sentences
To rewrite the irrational cube root ∛189 as a product of an integer and another irrational cube root, I need to find a perfect cube factor of 189. First, I'll find the prime factorization of 189, which is 3 × 3 × 3 × 7. Breaking it down further, I can see that 189 equals 27 × 7.
Now, I can rewrite the irrational cube root ∛189 as ∛(27 × 7). Since 27 is a perfect cube, I can simplify it as the cube root of 27, which is 3. Therefore, the irrational cube root ∛189 can be written as 3 (∛7).
To rewrite the irrational cube root ∛189 as a product of an integer and another irrational cube root, we need to find a perfect cube factor of 189.
Here's how we can do that:
Step 1: Prime factorize 189.
To prime factorize 189, we start by finding its prime factors. We can divide 189 by prime numbers until we cannot divide anymore.
189 ÷ 3 = 63
63 ÷ 3 = 21
21 ÷ 3 = 7
The prime factorization of 189 is: 3 × 3 × 7.
Step 2: Check for perfect cubes.
Now, we check if any of the prime factors are perfect cubes. In this case, we see that 3 is a perfect cube since 3 × 3 × 3 = 27.
Step 3: Rewrite the cube root.
Since we found a perfect cube factor, we can rewrite the cube root ∛189 as:
∛189 = ∛(27 × 7)
Step 4: Rearrange the terms.
Now, we can rearrange the terms to express the irrational cube root as a product of an integer and another irrational cube root:
∛(27 × 7) = ∛27 × ∛7
Step 5: Simplify the cube root of the perfect cube.
The cube root of 27 is 3, so we can simplify further:
∛27 × ∛7 = 3∛7
Therefore, the irrational cube root ∛189 can be rewritten as the product of an integer and another irrational cube root as 3∛7.