Video Transcript
Simplify four to the power of 𝑥
multiplied by four to the power of four 𝑥 divided by four to the power of three 𝑥
minus four multiplied by 16 to the power of 𝑥.
In order to simplify this
expression, we will need to use the laws of exponents. We recall that in order to use
these, each term must have the same base. In this question, three of the four
terms have a base of four. Our first step is therefore to
rewrite 16 to the power of 𝑥 with a base of four. We know that four squared is equal
to 16. This means that we can rewrite 16
to the power of 𝑥 as four squared to the power of 𝑥. The initial expression can
therefore be rewritten as four to the power of 𝑥 multiplied by four to the power of
four 𝑥 divided by four to the power of three 𝑥 minus four multiplied by four
squared to the power of 𝑥.
We will now consider three of the
laws of exponents. Firstly, the product rule states
that 𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the
power of 𝑚 plus 𝑛. This means that the numerator can
be rewritten as four to the power of 𝑥 plus four 𝑥. Before applying the same rule to
the denominator, we need to consider the power rule of exponents. This states that 𝑎 to the power of
𝑚 all raised to the power of 𝑛 is equal to 𝑎 to the power of 𝑚 multiplied by
𝑛. As such, four squared raised to the
power of 𝑥 can be rewritten as four to the power of two 𝑥.
We can then use the product rule to
rewrite the denominator as four to the power of three 𝑥 minus four plus two 𝑥. 𝑥 plus four 𝑥 is equal to five
𝑥, and three 𝑥 minus four plus two 𝑥 is equal to five 𝑥 minus four. So our expression simplifies to
four to the power of five 𝑥 divided by four to the power of five 𝑥 minus four.
We can now use the quotient rule,
which states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛 is equal to
𝑎 to the power of 𝑚 minus 𝑛. Subtracting five 𝑥 minus four from
five 𝑥, we have four to the power of five 𝑥 minus five 𝑥 minus four. Distributing the parentheses, the
exponent becomes five 𝑥 minus five 𝑥 plus four. And our expression simplifies to
four to the fourth power. Evaluating this, we get four
multiplied by four multiplied by four multiplied by four, which is equal to 256. Four to the power of 𝑥 multiplied
by four to the power of four 𝑥 divided by four to the power of three 𝑥 minus four
multiplied by 16 to the power of 𝑥 is equal to 256.
