How to Add 6 × 3^5 and 5 × 3^2 Using Exponentiation

Understanding exponentiation and how to incorporate it in arithmetic operations like addition is crucial for working with a variety of mathematical problems. This article will guide you through how to add 6 × 3^5 and 5 × 3^2 effectively.

Understanding Exponentiation

Exponentiation is a mathematical operation, written as an, involving two numbers: the base a and the exponent n. It represents repeated multiplication of the base by itself. That is, an a × a × ... × a (n times).

Breaking Down the Problem

The problem at hand is: 6×3^5 5×3^2. Let's start with breaking down each term.

The First Term: 6 × 3^5

Here, the base is 3, and the exponent is 5.

First, calculate 3^5:

[ 3^5 3 × 3 × 3 × 3 × 3 243 ]

Then, multiply by 6:

[ 6 × 243 1458 ]

The Second Term: 5 × 3^2

For the second term, the base is again 3, but the exponent is only 2.

First, calculate 3^2:

[ 3^2 3 × 3 9 ]

Then, multiply by 5:

[ 5 × 9 45 ]

Adding the Results

Now, we add the results of the first and second terms to get the final answer.

[ 1458 45 1503 ]

Other Methods to Approach the Problem

If you prefer a more algebraic approach or notice a common base, you can group like terms for simplification:

[ 6 × 3^5 5 × 3^2 ]

Note that 3^2 can be expressed as %3^2 3 × 3 9%, and 3^5 3^2 × 3^3:

[ 6 × 3^5 5 × 3^2 6 × 3^2 × 3^3 5 × 3^2 ]

Factor out the common term 3^2:

[ 3^2 (6 × 3^3 5) 9 (6 × 27 5) 9 (162 5) 9 × 167 1503 ]

Conclusion

By understanding and applying the rules of exponentiation and basic arithmetic operations, you can easily solve such problems. The key is to break down the exponents and handle them step by step.

Additional Resources

To further explore exponentiation and other mathematical concepts, check out online resources, math forums, or tutorials that cater to your specific needs and level of understanding.