Let's break down and solve the problems step by step:

The expressions for $A$ and $B$ are given as:

• $A = 3^4 \times 5^6$
• $B = 3^3 \times 5^2 \times 7$

Part (i): Find the value of the square root of $A$.

To find the square root of $A$, we need to take the square root of each prime factor raised to its power.

For $A = 3^4 \times 5^6$, we apply the square root to each factor:

$\sqrt{A} = \sqrt{3^4 \times 5^6} = \sqrt{3^4} \times \sqrt{5^6}$

$\sqrt{3^4} = 3^2 = 9 \quad \text{and} \quad \sqrt{5^6} = 5^3 = 125$

So,

$\sqrt{A} = 9 \times 125 = 1125$

Thus, the square root of $A$ is $1125$.

Part (ii): Find the highest common factor (HCF) of $A$ and $B$, leaving your answer in index notation.

To find the HCF, we take the lowest power of each common prime factor in the factorization of $A$ and $B$.

• For $3^4$ in $A$ and $3^3$ in $B$, the lowest power of 3 is $3^3$.
• For $5^6$ in $A$ and $5^2$ in $B$, the lowest power of 5 is $5^2$.
• The factor of 7 appears only in $B$, so it does not contribute to the HCF.

Therefore, the HCF is:

$HCF(A, B) = 3^3 \times 5^2$

Part (iii): Explain why $A$ is a perfect square.

For $A$ to be a perfect square, the power of each prime factor in its prime factorization must be even.

• In $A = 3^4 \times 5^6$, the powers of both 3 and 5 are even (4 and 6, respectively).
• Since all the prime factors of $A$ have even exponents, $A$ is a perfect square.

Final Answers:

• (i) The value of the square root of $A$ is $1125$.
• (ii) The HCF of $A$ and $B$ is $3^3 \times 5^2$.
• (iii) $A$ is a perfect square because all the exponents in its prime factorization are even.

Do you need further details on any of these steps? Let me know!

Here are 5 related questions to help deepen your understanding:

1. How do you determine if a number is a perfect square based on its prime factorization?
2. What is the relationship between the square root of a number and its prime factorization?
3. How do you find the least common multiple (LCM) of two numbers from their prime factorizations?
4. What would the HCF of $A = 3^4 \times 5^6 \times 7^2$ and $B = 3^3 \times 5^2 \times 7$ be?
5. How would you approach calculating the square root of $B$?

Tip: When working with prime factorization, always check if the exponents are even or odd. Even exponents mean the number is a perfect square!