𝑦 equals 𝑎 squared multiplied by 𝑏 to the power of 𝑥 plus two, where 𝑎 and 𝑏 are positive numbers. 𝑦 equals nine when 𝑥 equals negative two. 𝑦 equals 0.000576 when 𝑥 equals four. Work out the value of 𝑦 when 𝑥 equals five.
We’ve been given the relationship between 𝑥 and 𝑦 and we’re asked to work out the value of 𝑦 when 𝑥 is equal to five. But we don’t know the values of these numbers 𝑎 and 𝑏. So we’re going to need to determine them first. To do so, we need to use the other two pieces of information given in the question, which are two pairs of 𝑥- and 𝑦-values. We’ll be able to use these values to set up two equations which we can solve to find 𝑎 and 𝑏.
Using the first pair, we substitute nine for 𝑦 and negative two for 𝑥 giving nine equals 𝑎 squared multiplied by 𝑏 to the power of negative two plus two. Now, let’s see how this will simplify. In the power of 𝑏, we have negative two plus two which is just equal to zero. We then recall that anything to the power of zero is one. So we have 𝑎 squared multiplied by one which is just 𝑎 squared. Our equation is simplified to nine equals 𝑎 squared.
Now, we could go on and solve this equation for 𝑎 by square rooting both sides. But we don’t actually need to because the equation relating 𝑦 and 𝑥 actually uses 𝑎 squared. And we know that 𝑎 squared is equal to nine. So this is enough.
We can now use the second pair of values we were given to write a second equation. We get 0.000576 is equal to nine. Remember that’s 𝑎 squared multiplied by 𝑏 to the power of four plus two. We can simplify that power of 𝑏 because four plus two is just six. Our first step in solving this equation for 𝑏 is to divide both sides by nine leaving 𝑏 to the power of six on the right-hand side and giving 0.000064 on the left-hand side.
The final step in solving this equation is to take the sixth root of each side. And using a calculator to evaluate the sixth root of 0.000064 gives 0.2. Now technically, when taking a sixth root we should include plus or minus. Six is an even number. And so, both positive and negative 0.2 will give 0.000064 when raised to a power of six. However, we’re told in the question that both 𝑎 and 𝑏 are positive numbers. So we know that 𝑏 must be equal to the positive sixth root of 0.000064 which is 0.2.
So now that we know the values of 𝑎 squared and 𝑏, we can write down the relationship between 𝑥 and 𝑦 explicitly: 𝑦 is equal to nine multiplied by 0.2 to the power of 𝑥 plus two. We want to know the value of 𝑦 when 𝑥 is equal to five. So we need to make a substitution. We have that 𝑦 is equal to nine multiplied by 0.2 to the power of five plus two. That’s nine multiplied by 0.2 to the power of seven. And evaluating this on a calculator gives that 𝑦 is equal to 0.0001152.
Now, your calculator may actually have given the answer to this calculation in standard form. Mine did; it gave the answer as 1.152 multiplied by 10 to the power of negative four. To convert this number from standard form into a decimal, we’ll recall that because the power of 10 is negative, this means that the decimal needs to be getting smaller. And so, we’re going to be dividing 1.152 by 10 to the power of four.
To do so, we recall that to divide by 10 to the power of four, the decimal point must stay fixed and all of the digits need to move four places to the right. We can see that that’s what happened in our answer. The two has moved one, two, three, four places to the right and so have all of the other digits.
So we have our answer to the problem. We found that when 𝑥 is equal to five, 𝑦 is equal to 0.0001152.