Simplify 𝑥 times 𝑥 times 𝑥 times 𝑥 times 𝑥 times 𝑥 times 𝑥.
In this question, we are asked to simplify a product involving seven factors of 𝑥. There are a few different ways of simplifying this expression. And we will go through two of these.
The first way that we can simplify this expression is to recall how we define raising a number to a positive integer exponent. We can recall that if 𝑛 is a positive integer, then 𝑥 raised to the power of 𝑛 is the same as multiplying 𝑛 lots of 𝑥. In our expression, we can see that we have the product of seven lots of 𝑥. Therefore, this product must be equal to 𝑥 raised to the seventh power.
This is not the only way that we can simplify this product. We can note that 𝑥 is a monomial. So this expression is the product of monomials. We can then recall that we can simplify the product of monomials by using the product rule for exponents, which tells us that if 𝑚 and 𝑛 are nonnegative integers, then 𝑥 raised to the power of 𝑚 times 𝑥 raised to the power of 𝑛 is equal to 𝑥 raised to the power of 𝑚 plus 𝑛.
This result holds true for any number of factors in the product. And we can also recall that raising a number to the first power leaves it unchanged. So 𝑥 is equal to 𝑥 raised to the first power.
Therefore, we can apply the product rule for exponents to simplify this product. We obtain 𝑥 raised to the power of one plus one plus one plus one plus one plus one plus one. We can then evaluate the sum in the exponent to once again obtain an answer of 𝑥 raised to the seventh power.