Video Transcript
Given that two to the 𝑥 power equals four, determine the value of eight to the 𝑥 minus one power times two to the two 𝑥 plus six power times one-half to the two 𝑥 power.
We know that two to the 𝑥 power equals four. And we want to use that to solve this expression. In order for us to solve for an exponent variable, we rewrite this four with the base of two. We know that four is equal to two squared. And if two to the 𝑥 power equals two squared, then 𝑥 is equal to two. To solve the expression then, we’ll substitute two in every place we see 𝑥.
Eight to the 𝑥 minus one power becomes eight to the two minus one power. Two to the two 𝑥 plus six power is equal to two to the two times two plus six power. And then, we’ll have one-half to the two times two power. If we simplify that, we’ll have eight to the first power times two to the four plus six power, so two to the 10th power. And then, one-half to the two times two power, so one-half to the fourth power.
What we want to do now is distribute this four power across our fraction, which will be one to the fourth power over two to the fourth power. Eight to the first power equals eight. And then, we have two to the 10th power. We can rewrite this as eight times two to the 10th power because one to the fourth power just equals one. And then, put that numerator over the denominator of two to the fourth power. From there, we can cancel two to the fourth power and rewrite two to the 10th power as two to the sixth power.
We do this because of our exponent rules. When dividing exponents with the same base, we subtract the exponent values. Here, 10 minus four equals six. And then, we have eight times two to the sixth power. When we multiply two to the sixth power, we get 64. And eight times 64 equals 512. Given that two to the 𝑥 power equals four, the value of the expression we were given is 512.
