Simplify : `(216)^(3)xx (2500)^(2)xx (300)`

Answer

To simplify the expression \( (216)^3 \times (2500)^2 \times (300) \), we will break it down step by step.

### Step 1: Factor each number into its prime factors.

1. **Factor 216**:
- \( 216 = 2^3 \times 3^3 \) (since \( 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \))

2. **Factor 2500**:
- \( 2500 = 5^4 \times 2^2 \) (since \( 2500 = 25 \times 100 = (5^2) \times (10^2) = (5^2) \times (2 \times 5)^2 = 5^4 \times 2^2 \))

3. **Factor 300**:
- \( 300 = 3^1 \times 5^2 \times 2^2 \) (since \( 300 = 3 \times 100 = 3 \times (10^2) = 3 \times (2 \times 5)^2 = 3^1 \times 5^2 \times 2^2 \))

### Step 2: Rewrite the expression using the prime factorization.

Now we can rewrite the original expression using these factorizations:

\[ (216)^3 = (2^3 \times 3^3)^3 = 2^{3 \times 3} \times 3^{3 \times 3} = 2^9 \times 3^9 \]

\[ (2500)^2 = (5^4 \times 2^2)^2 = 5^{4 \times 2} \times 2^{2 \times 2} = 5^8 \times 2^4 \]

\[ 300 = 3^1 \times 5^2 \times 2^2 \]

### Step 3: Combine all the factors.

Now we can combine all the factors:

\[ (216)^3 \times (2500)^2 \times (300) = (2^9 \times 3^9) \times (5^8 \times 2^4) \times (3^1 \times 5^2 \times 2^2) \]

### Step 4: Group the same bases together.

Now, we group the same bases:

\[ = 2^{9 + 4 + 2} \times 3^{9 + 1} \times 5^{8 + 2} \]

### Step 5: Simplify the exponents.

Now we simplify the exponents:

\[ = 2^{15} \times 3^{10} \times 5^{10} \]

### Final Answer:

Thus, the simplified form of the expression \( (216)^3 \times (2500)^2 \times (300) \) is:

\[ 2^{15} \times 3^{10} \times 5^{10} \]

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