Arrange a,b,c in descending order if a=2^50*5^48*7^72 and b=2^52*5^47*7^73 and c=2^50*5^47*7^72

We are given three expressions for \( a \), \( b \), and \( c \), and asked to arrange them in descending order:

\[
a = 2^{50} \cdot 5^{48} \cdot 7^{72}
\]
\[
b = 2^{52} \cdot 5^{47} \cdot 7^{73}
\]
\[
c = 2^{50} \cdot 5^{47} \cdot 7^{72}
\]

### Step-by-step comparison of \( a \), \( b \), and \( c \):

1. **Comparing \( a \) and \( b \):**
   - \( a = 2^{50} \cdot 5^{48} \cdot 7^{72} \)
   - \( b = 2^{52} \cdot 5^{47} \cdot 7^{73} \)
   
   - Comparing the powers of \( 2 \):
     - \( a \) has \( 2^{50} \), and \( b \) has \( 2^{52} \). So, \( b \) has more factors of 2.
   - Comparing the powers of \( 5 \):
     - \( a \) has \( 5^{48} \), and \( b \) has \( 5^{47} \). So, \( a \) has more factors of 5.
   - Comparing the powers of \( 7 \):
     - \( a \) has \( 7^{72} \), and \( b \) has \( 7^{73} \). So, \( b \) has more factors of 7.

   Overall, \( b \) has more factors of 2 and 7, while \( a \) has more factors of 5. Since \( 7 \) has a higher weight (because of its larger power in both \( a \) and \( b \)), we can conclude that \( b > a \).

2. **Comparing \( a \) and \( c \):**
   - \( a = 2^{50} \cdot 5^{48} \cdot 7^{72} \)
   - \( c = 2^{50} \cdot 5^{47} \cdot 7^{72} \)
   
   The only difference between \( a \) and \( c \) is the power of 5, where \( a \) has \( 5^{48} \) and \( c \) has \( 5^{47} \). Since \( a \) has more factors of 5, \( a > c \).

3. **Comparing \( b \) and \( c \):**
   - \( b = 2^{52} \cdot 5^{47} \cdot 7^{73} \)
   - \( c = 2^{50} \cdot 5^{47} \cdot 7^{72} \)

   - Comparing the powers of \( 2 \):
     - \( b \) has \( 2^{52} \), and \( c \) has \( 2^{50} \). So, \( b \) has more factors of 2.
   - Comparing the powers of \( 5 \):
     - Both \( b \) and \( c \) have \( 5^{47} \), so they are equal in this aspect.
   - Comparing the powers of \( 7 \):
     - \( b \) has \( 7^{73} \), and \( c \) has \( 7^{72} \). So, \( b \) has more factors of 7.

   Since \( b \) has more factors of 2 and 7, we conclude that \( b > c \).

### Final conclusion:
The descending order of \( a \), \( b \), and \( c \) is:
\[
b > a > c
\]

Would you like further details or have any other questions?

Here are 5 related questions for practice:
1. If \( a = 2^{10} \cdot 3^5 \cdot 7^2 \) and \( b = 2^8 \cdot 3^6 \cdot 7^3 \), which is larger?
2. How would the order change if the powers of 5 were equal in all three terms?
3. What is the prime factorization of a number like \( 2^{10} \cdot 3^4 \cdot 5^6 \)?
4. How does the magnitude of exponents affect the overall comparison of large numbers?
5. How do you compare numbers of the form \( p^a \cdot q^b \cdot r^c \) when \( p, q, r \) are primes?

**Tip:** When comparing large products with different bases and exponents, prioritize the larger bases with higher exponents for quicker estimation of magnitude.

Arrange a, b, c in descending order if a = 2^50 * 5^48 * 7^72, b = 2^52 * 5^47 * 7^73, and c = 2^50 * 5^47 * 7^72.

Learn how to compare and arrange large numbers using prime factorization and the laws of exponents. This problem involves three expressions: a = 2^50 * 5^48 * 7^72, b = 2^52 * 5^47 * 7^73, and c = 2^50 * 5^47 * 7^72. Detailed step-by-step explanation provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation