Let's analyze and solve the given mathematical expression from the image.

The expression is:

$\frac{98^4 \times 21^3 \times 24^2 \times 4^3}{7^{10} \times 48^2 \times 12^3}.$

Step-by-step solution:

Step 1: Factorize the base numbers

We break down each number into its prime factorization:

• $98 = 2 \times 7^2$,
• $21 = 3 \times 7$,
• $24 = 2^3 \times 3$,
• $4 = 2^2$,
• $7 = 7$ (already prime),
• $48 = 2^4 \times 3$,
• $12 = 2^2 \times 3$.

Step 2: Substitute the factorizations into the expression

$\frac{(2 \cdot 7^2)^4 \cdot (3 \cdot 7)^3 \cdot (2^3 \cdot 3)^2 \cdot (2^2)^3}{7^{10} \cdot (2^4 \cdot 3)^2 \cdot (2^2 \cdot 3)^3}.$

Now simplify the powers for each term:

• $(2 \cdot 7^2)^4 = 2^4 \cdot 7^8$,
• $(3 \cdot 7)^3 = 3^3 \cdot 7^3$,
• $(2^3 \cdot 3)^2 = 2^6 \cdot 3^2$,
• $(2^2)^3 = 2^6$,
• $(2^4 \cdot 3)^2 = 2^8 \cdot 3^2$,
• $(2^2 \cdot 3)^3 = 2^6 \cdot 3^3$.

Step 3: Rewrite the entire expression

$\frac{(2^4 \cdot 7^8) \cdot (3^3 \cdot 7^3) \cdot (2^6 \cdot 3^2) \cdot 2^6}{7^{10} \cdot (2^8 \cdot 3^2) \cdot (2^6 \cdot 3^3)}.$

Combine the powers of each prime factor in the numerator and the denominator.

Step 4: Simplify the powers

Numerator:

• Powers of $2$: $2^4 \cdot 2^6 \cdot 2^6 = 2^{16}$,
• Powers of $7$: $7^8 \cdot 7^3 = 7^{11}$,
• Powers of $3$: $3^3 \cdot 3^2 = 3^5$.

So, the numerator becomes: $2^{16} \cdot 7^{11} \cdot 3^5.$

Denominator:

• Powers of $2$: $2^8 \cdot 2^6 = 2^{14}$,
• Powers of $7$: $7^{10}$,
• Powers of $3$: $3^2 \cdot 3^3 = 3^5$.

So, the denominator becomes: $2^{14} \cdot 7^{10} \cdot 3^5.$

Step 5: Simplify the fraction

• For $2^{16} / 2^{14} = 2^2$,
• For $7^{11} / 7^{10} = 7^1$,
• For $3^5 / 3^5 = 1$.

Thus, the simplified expression is: $2^2 \cdot 7 = 4 \cdot 7 = 28.$

Final Answer:

$\boxed{28}$

Would you like me to go into further detail or discuss related concepts?

5 Related Questions:

1. How do you approach simplifying expressions with exponents and prime factorizations?
2. What are the rules of exponents, and how do they apply to division and multiplication?
3. How do you handle fractional exponents or roots in similar problems?
4. Can you generalize this problem for variables instead of numbers?
5. How is prime factorization useful in real-world scenarios, like cryptography?

Tip:

When simplifying large expressions, always break down numbers into their prime factors to keep calculations systematic and manageable.