This image includes multiple math questions regarding prime factorization, perfect squares, and perfect cubes. Let's break down each part:

Part (a): Express 129,600 as a product of its prime factors.

• To solve this, we'll perform prime factorization on 129,600.

Part (b): Using the answer from part (a), explain why 129,600 is a perfect square.

• After the prime factorization, we'll analyze if each prime factor has an even exponent, which would confirm it is a perfect square.

Part (c): Given that $p$ and $q$ are prime numbers, find the values of $p$ and $q$ such that $\frac{129600p}{q}$ is a perfect cube.

• We'll need to adjust the prime factorization of 129,600 by multiplying it with $p$ and dividing by $q$ in such a way that the resulting expression is a perfect cube.

Let me first handle part (a) by finding the prime factorization of 129,600. sympy as sp

Finding the prime factorization of 129600

number = 129600 prime_factors = sp.factorint(number) prime_factorsThe prime factorization of 129,600 is:

$129,600 = 2^6 \times 3^4 \times 5^2$

For part (b): To check if 129,600 is a perfect square, we observe the exponents of the prime factors. For a number to be a perfect square, each prime factor's exponent must be even.

In this case:

• The exponent of $2$ is 6 (even),
• The exponent of $3$ is 4 (even),
• The exponent of $5$ is 2 (even).

Since all exponents are even, 129,600 is indeed a perfect square.

Now, let's move to part (c), where we need to find prime numbers $p$ and $q$ such that $\frac{129600p}{q}$ is a perfect cube. I'll calculate this next.To make $\frac{129600p}{q}$ a perfect cube, the exponents of all prime factors must be multiples of 3.

Currently, the prime factorization of 129,600 is $2^6 \times 3^4 \times 5^2$, with the following exponents:

• For $2$, we have $6$ (which is a multiple of 3, so no change needed).
• For $3$, we have $4$ (we need to add 2 to make it a multiple of 3).
• For $5$, we have $2$ (we need to add 1 to make it a multiple of 3).

Therefore, to make $\frac{129600p}{q}$ a perfect cube:

• We can multiply by $p = 3^2$ (to increase the exponent of 3 to 6),
• We can divide by $q = 5^1$ (to increase the exponent of 5 to 3).

Thus, $p = 9$ and $q = 5$.

Would you like more explanation or further details on any part of this process? Here are some related questions:

1. How do we determine the smallest prime factors of a number?
2. What is the definition of a perfect square and how can we identify one?
3. How can we generalize the process of making a number a perfect cube?
4. What happens if the exponents of prime factors are not adjusted to multiples of 3 in a perfect cube?
5. How does the prime factorization of a number affect its properties (like being a square or a cube)?

Tip: Always check if the exponents of prime factors are in the required form (even for squares, multiples of 3 for cubes) to quickly determine if a number fits a specific condition!