Video Transcript
Simplify root five raised to the
sixth power times root five squared all divided by root five raised to the fourth
power.
In this question, we are asked to
simplify an expression involving the product and quotient of exponential
expressions.
There are many ways of simplifying
this expression. We are going to start by noting
that the base in all of these exponential expressions is the same. So we will start by applying the
product rule for exponents in the numerator. This 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. In other words, when multiplying
exponential expressions with the same base, we can instead raise the base to the sum
of the exponents. This allows us to simplify the
numerator of the expression to obtain root five raised to the power of six plus two
over root five raised to the fourth power. We can then evaluate the exponent
to get root five raised to the eighth power over root five raised to the fourth
power.
We now have the quotient of two
exponential expressions with the same base. So we can simplify by using the
quotient rule for exponents. This tells us that 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 exponential expressions with the same base, we can instead raise the
base to the difference in the exponents. This holds true for any nonzero
base and integer exponents. Applying this to our expression
gives us root five raised to the power of eight minus four. We can then evaluate the exponent
to obtain root five raised to the fourth power.
We might be tempted to leave our
answer like this. However, we can simplify further by
recalling that if 𝑎 is nonnegative, the root of 𝑎 squared is equal to 𝑎. So, we can rewrite root five raised
to the fourth power as root five squared times root five squared. We can then evaluate each factor
separately to get five times five, which we can calculate is equal to 25.
