Write an exponential equation in the form of 𝑦 equals 𝑏 to the power of 𝑥 for the numbers in the table. 𝑥 equals two. 𝑦 equals nine sixteenths. 𝑥 equals four. 𝑦 equals 81 over 256. 𝑥 equals five. 𝑦 equals 243 over 1024.
We’ve already been told the form that our exponential equation should take. It is 𝑦 is equal to 𝑏 to the power of 𝑥. So what are we really looking to work out? We’re looking to find the value of 𝑏 for the set of numbers listed. And to do this, we can choose any pair of values in the table.
Let’s choose the first two. 𝑥 is equal to two. And 𝑦 is equal to nine sixteenths. We’re going to substitute these into the equation given. And at this point, it’s important to know that we could do this with any pair of numbers in the table. We’ve chosen two and nine sixteenths mainly because they’re such small numbers. 𝑦 is equal to nine sixteenths. And 𝑥 is equal to two. So our equation is nine sixteenths is equal to 𝑏 squared.
To solve this equation for 𝑏, we find the square root of both sides of the equation. The square root of 𝑏 is 𝑏. So we need to work out the square root of nine sixteenths. But remember, finding a square root yields a positive and a negative result. To square root nine sixteenths, we simply square root both the numerator and the denominator of the fraction. The square root of nine is three. And the square root of 16 is four. So 𝑏 could be plus or minus three-quarters.
So how do we decide whether it is a positive or negative? Well, this time, we’re going to consider the third entry in the table. 𝑥 is equal to five when 𝑦 is equal to 243 over 1024. Let’s use the positive value of 𝑏 first. And we’ll substitute 𝑥 is equal to five into this equation. That’s three-quarters to the power of five, which gives us 243 over 1024, which is what we we’re expecting. So this works for a positive value of three-quarters.
Let’s repeat this process with negative three-quarters. This time, we have negative three-quarters to the power of five. That gives us negative 243 over 1024. That’s not the answer we’re looking for. So we can say that 𝑏 must be three-quarters. And therefore, the equation in exponential form is 𝑦 is equal to three-quarters to the power of 𝑥.
And in fact, we can perform one final check. This time, we’ll substitute the as-yet unused values in the second entry in our table. That’s 𝑥 is equal to four. And 𝑦 is equal to 81 over 256. 𝑦 is equal to three-quarters to the power of four, which is indeed 81 over 256. And 𝑦 is equal to three-quarters to the power of 𝑥.