Question Video: Finding an Unknown by Comparing Powers of Real Numbers Mathematics

Video Transcript

Find the value of 𝑥 in the equation two to the 𝑥 minus three power equals root two to the 𝑥 plus one power.

To find the value of 𝑥 in this equation, we can use the rule 𝑎 to the 𝑚 power equals 𝑎 to the 𝑛 power, where 𝑎 is in the set of real numbers without negative one, zero, or one, then 𝑚 is equal to 𝑛. However, this rule only works when both sides of the equation have the same base. The left side of our equation has base two and the right side base root two. So let’s aim to rewrite the right side with a base of two also.

One rule that we can recall here is that we can write the square root of 𝑎 as 𝑎 to the half power. So root two can be written as two to the half power. So our original equation can be written as two to the 𝑥 minus three power equals two to the half power to the 𝑥 plus one power.

Let’s see if we can simplify this right-hand side of the equation. To help us do this, let’s recall how we can raise a number to a power to another power. We use the rule 𝑥 to the 𝑎 power raised to the 𝑏 power is 𝑥 to the 𝑎𝑏 power. Therefore, two to the half power raised to the 𝑥 plus one power is two to the half multiplied by 𝑥 plus one power. Since the bases are now the same, we can equate the exponents using the rule that we wrote down here. 𝑥 minus three equals half multiplied by 𝑥 plus one.

We can begin to solve this by multiplying both sides of the equation by two, then adding six to both sides, and finally subtracting 𝑥 from both sides, to give us 𝑥 equals seven. Therefore, the value of 𝑥 for this equation is seven.