Finding the Number of Prime Factors in a Product

To find the number of prime factors in the product of two numbers raised to a power, we first need to find the prime factorization of each base number. Then, we apply the rules of exponents and combine the prime factors. Finally, we sum the exponents of the prime factors in the resulting expression to get the total count of prime factors.

Prime Factorization of 30

We start by finding the prime factors of the number 30.

$$30 = 2 \times 15$$ $$30 = 2 \times 3 \times 5$$

The prime factorization of 30 is $2^1 \times 3^1 \times 5^1$. The prime factors are 2, 3, and 5.

Prime Factorization of 24

Next, we find the prime factors of the number 24.

$$24 = 2 \times 12$$ $$24 = 2 \times 2 \times 6$$ $$24 = 2 \times 2 \times 2 \times 3$$ $$24 = 2^3 \times 3^1$$

The prime factorization of 24 is $2^3 \times 3^1$. The prime factors are 2 and 3.

Expressing the Product (30)⁵ × (24)⁵ Using Prime Factors

Now, we will express each term in the product $(30)^5 \times (24)^5$ using their prime factorizations and apply the power of 5.

For $(30)^5$:

$$(30)^5 = (2 \times 3 \times 5)^5$$ Using the rule $(abc)^m = a^m b^m c^m$, we get: $$(2 \times 3 \times 5)^5 = 2^5 \times 3^5 \times 5^5$$

For $(24)^5$:

$$(24)^5 = (2^3 \times 3)^5$$ Using the rule $(a^n b^m)^p = a^{np} b^{mp}$, we get: $$(2^3 \times 3)^5 = (2^3)^5 \times (3^1)^5$$ $$(2^3)^5 \times (3^1)^5 = 2^{3 \times 5} \times 3^{1 \times 5}$$ $$2^{15} \times 3^5$$

Now, we multiply the prime factorizations of $(30)^5$ and $(24)^5$:

$$(30)^5 \times (24)^5 = (2^5 \times 3^5 \times 5^5) \times (2^{15} \times 3^5)$$

Group terms with the same base:

$$(2^5 \times 2^{15}) \times (3^5 \times 3^5) \times 5^5$$

Using the rule $a^m \times a^n = a^{m+n}$, we add the exponents for each prime base:

$$2^{5+15} \times 3^{5+5} \times 5^5$$ $$2^{20} \times 3^{10} \times 5^5$$

Counting the Total Number of Prime Factors

The expression $2^{20} \times 3^{10} \times 5^5$ represents the prime factorization of the product $(30)^5 \times (24)^5$.

This means:

• The prime factor 2 appears 20 times.
• The prime factor 3 appears 10 times.
• The prime factor 5 appears 5 times.

To find the total number of prime factors, we sum the exponents:

Total number of prime factors = $20 + 10 + 5 = 35$.

Revision Table: Key Concepts

Additional Information on Prime Factorization and Exponents

Understanding prime factorization is fundamental in number theory. Every integer greater than 1 has a unique prime factorization, according to the Fundamental Theorem of Arithmetic. This theorem is crucial for many mathematical concepts, including finding the greatest common divisor (GCD) and the least common multiple (LCM) of numbers.

Exponents provide a shorthand way to write repeated multiplication. The rules of exponents are essential for simplifying expressions involving powers, especially when dealing with prime factorizations of large numbers or products, as seen in this problem. These rules help us efficiently combine terms with the same base.

For instance, calculating $(30)^5$ directly is complex, but using prime factors $(2 \times 3 \times 5)^5$ and applying exponent rules simplifies the process significantly to $2^5 \times 3^5 \times 5^5$. Similarly, multiplying $(30)^5$ and $(24)^5$ is made manageable by working with their prime factor forms and combining the exponents of the like bases.