Multiplying and dividing index laws

Multiplying Numbers with Indices

When you multiply numbers that both have indices, you add the indices together. This rule applies specifically when the base numbers (the large numbers) are the same.

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For example, consider the equation 2^2 \times 2^3:

  • First, identify that the base number, 2, is the same for both terms
  • Next, add the indices together: 2 + 3 = 5
  • The base number remains 2, and the new index is 5

Therefore, 2^2 \times 2^3 = 2^5.

Dividing Numbers with Indices

Similarly, when you divide numbers with indices, you subtract the indices. This rule is only applies when the base numbers are the same.

For example, to solve 3^5 \div 3^3:

  • Identify that the base number, 3, is the same for both terms
  • Subtract the indices: 5 - 3 = 2
  • The base number remains 3, and the new index is 2

Therefore, 3^5 \div 3^3 = 3^2.

Practice Questions

Question 1:

What is 4^3 \times 4^2?

Solution

The answer is 4^5.

Add the indices together: 3 + 2 = 5.

Question 2:

What is 5^6 \div 5^3?

Solution

The answer is 5^3.

Subtract the indices: 6 - 3 = 3.

Question 3:

What is 6^4 \times 6^3 \div 6^5?

Solution

The answer is 6^2.

First, add the indices for multiplication 4 + 3 = 7.

Then, subtract the index for division 7 - 5 = 2.

Question 4:

If a^m \times a^n = a^{10} and m = 3, what is n?

Solution

The answer is n = 7.

Since m + n = 10 and m = 3, we find n = 10 - 3 = 7.

 

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