I'll start by solving the first few problems from the image for you. These are square root simplifications involving variables and exponents.

1. $\sqrt{40a^4b^3c^2}$

We simplify this square root as follows:

• $\sqrt{40}$ simplifies to $2\sqrt{10}$ because $40 = 4 \times 10$ and $\sqrt{4} = 2$.
• For $a^4$, $\sqrt{a^4} = a^2$.
• For $b^3$, $\sqrt{b^3} = b\sqrt{b}$.
• For $c^2$, $\sqrt{c^2} = c$.

Thus, $\sqrt{40a^4b^3c^2} = 2a^2bc\sqrt{10b}$.

2. $\sqrt{49a^2b^5}$

• $\sqrt{49} = 7$.
• $\sqrt{a^2} = a$.
• $\sqrt{b^5} = b^2\sqrt{b}$.

So, $\sqrt{49a^2b^5} = 7ab^2\sqrt{b}$.

3. $\sqrt{121p^4q^9r^2}$

• $\sqrt{121} = 11$.
• $\sqrt{p^4} = p^2$.
• $\sqrt{q^9} = q^4\sqrt{q}$.
• $\sqrt{r^2} = r$.

Thus, $\sqrt{121p^4q^9r^2} = 11p^2q^4r\sqrt{q}$.

4. $\sqrt{36a^2b^5c^2}$

• $\sqrt{36} = 6$.
• $\sqrt{a^2} = a$.
• $\sqrt{b^5} = b^2\sqrt{b}$.
• $\sqrt{c^2} = c$.

So, $\sqrt{36a^2b^5c^2} = 6ab^2c\sqrt{b}$.

5. $\sqrt{144a^6b^2c^2}$

• $\sqrt{144} = 12$.
• $\sqrt{a^6} = a^3$.
• $\sqrt{b^2} = b$.
• $\sqrt{c^2} = c$.

Thus, $\sqrt{144a^6b^2c^2} = 12a^3bc$.

Would you like to proceed with the remaining problems or any specific one? Let me know if you want more details on the steps!

Here are 5 questions related to this topic:

1. How do you simplify square roots with variables and exponents?
2. What is the general rule for simplifying $\sqrt{a^n}$ when $n$ is even or odd?
3. How can you apply prime factorization to simplify square roots of larger numbers?
4. What is the process for simplifying cube roots compared to square roots?
5. How would you simplify expressions involving both square roots and fractions?

Tip:

When simplifying square roots of variables with exponents, remember that the square root of an even power is always straightforward, while odd powers will leave a factor under the square root.