Question

30 times the cube root of 1 - 10 times the cube root of 3

10 times the cube root of 81 - 10 times the cube root of 3

5 times the cube root of 9 + 10 times the cube root of 3

Answer 1

To simplify the expressions you've provided, we'll calculate each term step by step. We will also express cube roots clearly.

**Expression 1:**
\[
30 \cdot \sqrt[3]{1} - 10 \cdot \sqrt[3]{3}
\]

Since \(\sqrt[3]{1} = 1\):
\[
30 \cdot 1 - 10 \cdot \sqrt[3]{3} = 30 - 10 \cdot \sqrt[3]{3}
\]

**Expression 2:**
\[
10 \cdot \sqrt[3]{81} - 10 \cdot \sqrt[3]{3}
\]

Since \(\sqrt[3]{81} = \sqrt[3]{3^4} = 3 \cdot \sqrt[3]{3}\):
\[
10 \cdot (3 \cdot \sqrt[3]{3}) - 10 \cdot \sqrt[3]{3} = 30 \cdot \sqrt[3]{3} - 10 \cdot \sqrt[3]{3} = 20 \cdot \sqrt[3]{3}
\]

**Expression 3:**
\[
5 \cdot \sqrt[3]{9} + 10 \cdot \sqrt[3]{3}
\]

Since \(\sqrt[3]{9} = \sqrt[3]{3^2}\):
\[
5 \cdot \sqrt[3]{3^2} + 10 \cdot \sqrt[3]{3} = 5 \cdot \sqrt[3]{3^2} + 10 \cdot \sqrt[3]{3}
\]

Because \(\sqrt[3]{3^2} = \sqrt[3]{3}^2\):
Let \(x = \sqrt[3]{3}\), then this becomes:
\[
5x^2 + 10x
\]

Now you have three simplified expressions:

1. \(30 - 10\sqrt[3]{3}\)
2. \(20\sqrt[3]{3}\)
3. \(5\sqrt[3]{9} + 10\sqrt[3]{3}\) (or \(5\sqrt[3]{3^2} + 10\sqrt[3]{3}\)) which can be expressed as \(5\sqrt[3]{3}^2 + 10\sqrt[3]{3}\).

Let me know if you need further assistance!