Let's simplify the expression step by step:

$\sqrt{27} + 5\sqrt{3} - \sqrt[3]{75}$

Step 1: Simplify $\sqrt{27}$

The square root of 27 can be simplified by factoring 27 as $9 \times 3$, where 9 is a perfect square.

$\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$

Step 2: Simplify $\sqrt[3]{75}$

To simplify $\sqrt[3]{75}$, first factor 75 as $25 \times 3$, where 25 is a perfect cube of $5^2$.

[ \sqrt[3]{75} = \sqrt[3]{25 \times 3} = \sqrt[3]{25} \times \sqrt[3]{3} ]

Since 25 is not a perfect cube, the cube root of 25 remains as $\sqrt[3]{25}$, and the cube root of 3 remains as $\sqrt[3]{3}$.

Thus, $\sqrt[3]{75} = \sqrt[3]{25} \times \sqrt[3]{3}$, so it remains in its cube root form.

Step 3: Combine the expression

Now we combine everything back into the original expression:

[ \sqrt{27} + 5\sqrt{3} - \sqrt[3]{75} = 3\sqrt{3} + 5\sqrt{3} - \sqrt[3]{75} ]

We can combine the like terms $3\sqrt{3}$ and $5\sqrt{3}$:

[ (3 + 5)\sqrt{3} - \sqrt[3]{75} = 8\sqrt{3} - \sqrt[3]{75} ]

Final Answer:

$8\sqrt{3} - \sqrt[3]{75}$

This is the simplified form of the expression.

Would you like more details on any of the steps? Or have additional questions?

5 Related Questions:

1. How do you simplify cube roots when the number inside is not a perfect cube?
2. How can you simplify the cube root of a product?
3. What’s the difference between simplifying square roots and cube roots?
4. How do you add and subtract square roots and cube roots?
5. How do you handle radicals when they are mixed (i.e., square roots and cube roots together)?

Tip: When simplifying square or cube roots, always look for perfect squares or cubes inside the radical to make the process easier!