\sqrt{24b^3}\cdot\sqrt{40b^2}\cdot\sqrt{b^2}=

We need to simplify the expression:

\[
\sqrt{24b^3} \cdot \sqrt{40b^2} \cdot \sqrt{b^2}
\]

Step 1: Combine the square roots into one.

Since all terms are under square roots, we can combine them as follows:

\[
\sqrt{24b^3 \cdot 40b^2 \cdot b^2}
\]

Step 2: Simplify the multiplication inside the square root.

First, multiply the constants:

\[
24 \cdot 40 = 960
\]

Now, multiply the \(b\)-terms using the property \(b^a \cdot b^b = b^{a+b}\):

\[
b^3 \cdot b^2 \cdot b^2 = b^{3+2+2} = b^7
\]

Now the expression inside the square root becomes:

\[
\sqrt{960b^7}
\]

Step 3: Simplify the square root.

We can split this square root into two parts:

\[
\sqrt{960} \cdot \sqrt{b^7}
\]

Step 4: Simplify each square root.

- For \( \sqrt{960} \), we can simplify it by finding the largest perfect square factor of 960. The number 960 can be factored as \(960 = 64 \cdot 15\), so:

\[
\sqrt{960} = \sqrt{64 \cdot 15} = \sqrt{64} \cdot \sqrt{15} = 8\sqrt{15}
\]

- For \( \sqrt{b^7} \), we use the property that \( \sqrt{b^7} = b^{\frac{7}{2}} = b^3 \cdot \sqrt{b} \).

Step 5: Combine everything.

Now we combine the simplified parts:

\[
8\sqrt{15} \cdot b^3 \cdot \sqrt{b} = 8b^3 \cdot \sqrt{15b}
\]

Thus, the simplified expression is:

\[
8b^3\sqrt{15b}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the result change if one of the terms inside the square root had a different power of \(b\)?
2. What happens if you simplify \(\sqrt{b^5} \cdot \sqrt{b^3}\)?
3. How can you generalize this approach to any square roots of powers of variables?
4. Can you solve a similar problem where instead of square roots, cube roots are involved?
5. What happens if the coefficient inside the square root is a fraction?

**Tip**: When simplifying square roots of variables with exponents, separate the exponents into even and odd parts to easily manage square roots.

Simplify the radical expression \sqrt{24b^3}\cdot\sqrt{40b^2}\cdot\sqrt{b^2} step by step. This detailed solution covers exponent laws, multiplication of radicals, and simplifying square roots. Suitable for students in Grades 9-10.

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