Video Transcript
Simplify 𝑥 raised to the seventh
power divided by 𝑥 raised to the sixth power.
In this question, we are asked to
simplify the quotient of a variable 𝑥 raised to two different exponents.
Since we are asked to simplify the
quotient of two exponential expressions with the same base, we can start by
recalling that the quotient rule for exponents tells us for a nonzero base 𝑏, 𝑏
raised to the power of 𝑚 over 𝑏 raised to the power of 𝑛 is equal to 𝑏 raised to
the power of 𝑚 minus 𝑛. In other words, when taking the
quotient of two exponential expressions with the same base, we can instead raise the
base to the difference in the exponents.
To apply this to the given
expression, we have 𝑏 equals 𝑥, 𝑚 equals seven, and 𝑛 equals six. So we have 𝑥 raised to the power
of seven minus six, provided 𝑥 is nonzero. We can then evaluate the expression
in the exponent to obtain 𝑥 raised to the first power.
Finally, we can recall that raising
any number to an exponent of one leaves it unchanged, so 𝑥 raised to the first
power is equal to 𝑥. Hence, we can simplify 𝑥 raised to
the seventh power over 𝑥 raised to the sixth power to be 𝑥 provided that 𝑥 is
nonzero.
