Simplify 𝑥 to the seventh times 𝑥 to the negative fifth times 𝑥 to the fourth, where 𝑥 is not equal to zero.
We can use the rule where 𝑥 to the 𝑎 times 𝑥 to the 𝑏 is equal to 𝑥 to the 𝑎 plus 𝑏. So when we multiply things with like bases, we add their exponents. So we need to take this, keep our like base cause they’ll have a base of 𝑥, and then add our exponents: seven plus negative five plus four, which gives us 𝑥 to the sixth.
And we get 𝑥 to the sixth. Now there is another way to do this. It may take a little bit longer, but let’s give it a try. So in the original problem, we have 𝑥 to the negative fifth power. When you have a negative exponent and it’s on a numerator, we move it to the denominator and make it positive.
Or vice versa, if it was a negative exponent on the denominator, we can move it up to the numerator. So what we can do, keep 𝑥 to the seventh and 𝑥 to the fourth on the numerator but move the 𝑥 to the negative fifth. So just like we did before, we need to add the seven and four together.
And seven plus four, and we get 𝑥 to the 11th. Now when we divide, we need to subtract our exponents. And 11 minus 5 gives us 𝑥 to the sixth. So once again 𝑥 to the sixth is our final answer.