Simplify the following expression, writing the answer using index notation: 151115315 3 15 11 ​

To simplify the expression $151115315 \times 3 \times 15 \times 11$ using index notation, we will follow these steps:

1. ### Break Down the Problem

1. Identify and group the terms in the expression.
2. Write the numeric values in index notation (prime factorization).

2. ### Relevant Concepts

1. We can represent each number as a product of its prime factors.
2. Use index notation to express repeated factors.

3. ### Analysis and Detail

1. Prime Factorization:

   • $15 = 3 \times 5 = 3^1 \times 5^1$
   • $11 = 11^1$
   • $3 = 3^1$
   • $151115315 = 15 \times 11 \times 15 \times 3 \times 15$
   • Breaking down $151115315$:
     • $15: (3^1 \times 5^1)^3$, since there are three instances of 15.
     • $11: 11^1$
     • $3: 3^1$

2. Combine the factors:

   • Count the exponents of each prime factor:
     • For $3$: There are three $15's$ contributing $3 \times 3^1 = 3^3$ from $15$ and one additional $3$, so $3^3 + 3^1 = 3^4$.
     • For $5$: From three $15's$, $5^1 \times 3 = 5^3$.
     • For $11$: There is one $11$, so $11^1$.

3. Putting it together:

   • Therefore, combining yields: $151115315 \times 3 \times 15 \times 11 = 3^4 \times 5^3 \times 11^1$

4. ### Verify and Summarize

• Ensure all factors and indices were added correctly.
• The expression is now expressed in index notation.

Final Answer

The simplified expression in index notation is: $3^4 \times 5^3 \times 11^1$