Write an exponential equation in the form 𝑦 is equal to 𝑎 multiplied by 𝑏 to the power of 𝑥 for the numbers in the table. We are given four pairs of values. When 𝑥 equals zero, 𝑦 is equal to five; when 𝑥 equals one, 𝑦 is 15; when 𝑥 is two, 𝑦 is 45; and when 𝑥 is equal to three, 𝑦 is equal to 135.
In order to calculate the values of the constants 𝑎 and 𝑏, we need to substitute the values of 𝑥 and 𝑦 into the equation. Substituting 𝑥 equals zero and 𝑦 equals five gives us five is equal to 𝑎 multiplied by 𝑏 to the power of zero. We know that anything to the power of zero is equal to one. This means that the constant 𝑎 is equal to five. Substituting 𝑥 equals one and 𝑦 equals 15 into the equation gives us 15 is equal to 𝑎 multiplied by 𝑏 to the power of one.
Anything to the power of one is itself, and we already know that 𝑎 equals five. This means that 15 is equal to five 𝑏. Dividing both sides of this equation by five gives us 𝑏 is equal to three. We now have values of 𝑎 and 𝑏 that we can substitute into the exponential equation. This gives us an answer of 𝑦 is equal to five multiplied by three to the power of 𝑥.
At this stage, it is worth checking that this equation works for the third and fourth pair of values. When 𝑥 is equal to two and 𝑦 is equal to 45, we have 45 is equal to five multiplied by three squared. Three squared is equal to nine. Five multiplied by nine is equal to 45. For our final pair of values, we have 135 is equal to five multiplied by three cubed. Three cubed is 27. Multiplying this by five give us 135. The equation also works for this pair of values. This confirms that for the numbers in the table, the equation in exponential form is 𝑦 is equal to five multiplied by three to the power of 𝑥.