### Video Transcript

Simplify 𝑏 raised to the fifth
power times 𝑏 squared.

In this question, we are asked to
simplify a product. We can note that each factor is an
exponential expression with the same base. We can then recall that we can
simplify the product of exponential expressions by using the product rule for
exponents, which tells us that 𝑏 raised to the power of 𝑚 times 𝑏 raised to the
power of 𝑛 is equal to 𝑏 raised to the power of 𝑚 plus 𝑛, for any nonnegative
integers 𝑚 and 𝑛.

In other words, we can find the
product of these exponential expressions with the same base by raising the base 𝑏
to the sum of the exponents. We obtain 𝑏 raised to the power of
five plus two. We can then evaluate the expression
in the exponent to obtain 𝑏 raised to the seventh power.

While this is enough to answer the
question, it can be useful to show this calculation in full to show why this result
is actually true. We can do this by recalling that
raising a number to a positive integer exponent 𝑛 is the same as multiplying 𝑛
lots of that number together. Therefore, 𝑏 raised to the power
of five is 𝑏 times 𝑏 times 𝑏 times 𝑏 times 𝑏, and 𝑏 squared is 𝑏 times
𝑏. We see that the product of these
expressions can be thought of as the product of five plus two lots of 𝑏.

Therefore, we can write this
product as 𝑏 raised to the power of five plus two, which is equal to 𝑏 raised to
the seventh power. This outlines the proof of the
product rule for exponents with positive integer exponents. However, we have shown that 𝑏
raised to the fifth power times 𝑏 squared is equal to 𝑏 raised to the seventh
power.