Question

Fine the cube root of -729: 3√-729
Answer
find the cube root of -8: 3√-8
Answer
find the cubic root of 135
Answer
find the cubic root of 135
5 cubic root 3√27
3 cubic root 3√5
3√5
cubic root 3√5
Answer
what is the circumference of a circle with a diameter of 4.5 miles? estimate the circumference to the hundredths place or using 3.14 for pi
Answer
rewrite the irrational cube root 3√189 as the product of an integer and another irrational cube root. show your work.
Answered by GPT-4o mini
The cube root of -729 can be calculated as follows:

\[
\sqrt[3]{-729} = -\sqrt[3]{729}
\]

Since \(729\) is \(9^3\) (because \(9 \times 9 \times 9 = 729\)), we have:

\[
\sqrt[3]{729} = 9
\]

Therefore,

\[
\sqrt[3]{-729} = -9
\]

So, the cube root of -729 is \(-9\).
Answered by GPT-4o mini
To find the cube root of \(-8\), we can write it as:

\[
\sqrt[3]{-8} = \sqrt[3]{-2^3}
\]

Since the cube root of a negative number is also negative, we have:

\[
\sqrt[3]{-2^3} = -2
\]

Therefore, the cube root of \(-8\) is \(-2\).
Answered by GPT-4o mini
To find the cube root of \(135\), you would typically use the notation:

\[
\sqrt[3]{135}
\]

The cube root of \(135\) can be approximated, as \(135\) is not a perfect cube. You can calculate it using a calculator, or estimate it by finding two nearby perfect cubes.

The perfect cubes near \(135\) are \(125\) (which is \(5^3\)) and \(216\) (which is \(6^3\)). Since \(135\) is between \(125\) and \(216\), we know that:

\[
5 < \sqrt[3]{135} < 6
\]

To get a more precise value, you can use a calculator:

\[
\sqrt[3]{135} \approx 5.13
\]

Thus, the approximate value of the cube root of \(135\) is about \(5.13\).
Answered by GPT-4o mini
It looks like you're asking for the cube root of \(135\) in a simplified form. Let's break this down step by step.

1. First, we know that \(135\) can be factored into prime factors:

\[
135 = 3^3 \times 5
\]

2. Now, we can find the cube root of \(135\):

\[
\sqrt[3]{135} = \sqrt[3]{3^3 \times 5}
\]

3. Since the cube root of \(3^3\) is \(3\), we have:

\[
\sqrt[3]{135} = 3 \times \sqrt[3]{5}
\]

So, the cube root of \(135\) can be expressed as:

\[
\sqrt[3]{135} = 3 \sqrt[3]{5}
\]

To summarize,

\[
\sqrt[3]{135} = 3 \sqrt[3]{5}
\]

If you were referring to something else or need another specific form, please clarify!
Answered by GPT-4o mini
To calculate the circumference of a circle, you can use the formula:

\[
C = \pi \times d
\]

where \(C\) is the circumference and \(d\) is the diameter.

Given that the diameter is \(4.5\) miles and using \(\pi \approx 3.14\):

\[
C = 3.14 \times 4.5
\]

Now, perform the multiplication:

\[
C \approx 3.14 \times 4.5 = 14.13 \text{ miles}
\]

So, the estimated circumference of a circle with a diameter of \(4.5\) miles is approximately **14.13 miles**.
Answered by GPT-4o mini
To rewrite the irrational cube root \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we first need to factor \(189\) into its prime factors.

1. **Factor \(189\)**:
- \(189\) can be divided by \(3\):
\[
189 \div 3 = 63
\]
- Then, factor \(63\) by dividing by \(3\) again:
\[
63 \div 3 = 21
\]
- Finally, factor \(21\) by dividing by \(3\):
\[
21 \div 3 = 7
\]
- The complete factorization of \(189\) is:
\[
189 = 3^3 \times 7
\]

2. **Express \(\sqrt[3]{189}\)**:
\[
\sqrt[3]{189} = \sqrt[3]{3^3 \times 7}
\]

3. **Apply the property of cube roots**:
- We can separate the expression using the property of cube roots:
\[
\sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7}
\]

4. **Simplify the cube root**:
- Since \(\sqrt[3]{3^3} = 3\), we get:
\[
\sqrt[3]{189} = 3 \times \sqrt[3]{7}
\]

Thus, the expression \(\sqrt[3]{189}\) can be rewritten as:

\[
\sqrt[3]{189} = 3\sqrt[3]{7}
\]

This shows \(\sqrt[3]{189}\) as the product of an integer (\(3\)) and another irrational cube root (\(\sqrt[3]{7}\)).