Write the following expression using exponents: eleven times eleven times 𝑎 times 𝑎 times 𝑎 times 𝑎 times 𝑎. Looking at what we have, there are two common factors of eleven and five common factors of 𝑎. Now using the associative property, we can add in some parentheses.
If you are adding or multiplying, you can use the associative property to group certain numbers together and you can actually pick the order. So for this case we’ve grouped together the two common factors of eleven and the five common factors of 𝑎.
By the definition of exponents, we have eleven to the second power times 𝑎 to the fifth power. We can write this because eleven is our base and the base should be our common factor, and our common factor is eleven. Two is our exponent, and our exponent tells us how many times the base is used as a factor. Well we had two elevens, so our exponent should be two.
Using the definition of exponents again, 𝑎 is our base because 𝑎 was our common factor, and five is our exponent because our common factor 𝑎 was used five times. Therefore, our final answer is eleven to the second power times 𝑎 to the fifth Power. So we took our expression and we rewrote it using exponents.