Video Transcript
Find the value of 𝑎 for which two
to the power of 𝑥 plus six minus two to the power of 𝑥 plus two is equal to 𝑎
multiplied by two to the power of 𝑥.
In order to find the value of 𝑎 in
this equation, we need the left-hand side of the equation to be in the form of the
right-hand side. In other words, we need to rewrite
two to the power of 𝑥 plus six minus two to the power of 𝑥 plus two so it is in
the form 𝑎 multiplied by two to the power of 𝑥.
We recall that the product rule of
exponents 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 we can rewrite the
first term as two to the power of 𝑥 multiplied by two to the power of six. Likewise, two to the power of 𝑥
plus two can be rewritten as two to the power of 𝑥 multiplied by two squared. Both of our terms now have a common
factor of two to the power of 𝑥.
Factoring this out, we have two to
the power of 𝑥 multiplied by two to the sixth power minus two squared. We know that two squared is equal
to four and two to the sixth power is 64. As such, our expression simplifies
to two to the power of 𝑥 multiplied by 60 or 60 multiplied by two to the power of
𝑥. Since this is equal to 𝑎
multiplied by two to the power of 𝑥, we can compare the coefficients, giving us a
value of 𝑎 equal to 60.
