Write an exponential equation in the form 𝑦 equals 𝑎 multiplied by 𝑏 to the power of 𝑥 for the numbers in the table. We’re given in the table that when 𝑥 equals zero, 𝑦 equals 18. When 𝑥 equals one, 𝑦 equals six. When 𝑥 equals two, 𝑦 is equal to two. And when 𝑥 is equal to three, 𝑦 is equal to two-thirds.
In order to answer this question, we need to work out the values of the constants 𝑎 and 𝑏. We can do this by substituting in the values of 𝑥 and 𝑦 from the table. Let’s consider the first column, 𝑥 equals zero and 𝑦 equals 18. Substituting in these values gives us 18 is equal to 𝑎 multiplied by 𝑏 to the power of zero. We can call this equation one. Substituting in the values in the second column, 𝑥 equals one and 𝑦 equals six, gives us six is equal to 𝑎 multiplied by 𝑏 to the power of one. We will call this equation two.
We now need to remember two of our laws of exponents or indices. Firstly, anything to the power of zero is equal to one. Secondly, anything to the power of one is equal to itself. 𝑥 to the power of zero equals one. And 𝑥 to the power of one equals 𝑥. This means that, in our equations, 𝑏 to the power of zero is also equal to one. And 𝑏 to the power of one can be written as 𝑏. Equation one simplifies to 18 is equal to 𝑎 multiplied by one. Therefore, 𝑎 is equal to 18. Equation two simplifies to six is equal to 𝑎 multiplied by 𝑏.
We have just worked out that 𝑎 is equal to 18. Therefore, six is equal to 18𝑏. Dividing both sides of this equation by 18 gives us six over 18 is equal to 𝑏. The fraction six over 18 can be simplified to one-third by dividing the numerator and denominator by six. Therefore, 𝑏 is equal to one-third. We can now substitute the values of 𝑎 and 𝑏 into our exponential equation. The equation becomes 𝑦 is equal to 18 multiplied by one-third to the power of 𝑥.
We could check this equation by substituting in the values in the third and fourth column, for example, when 𝑥 equals two, 𝑦 equals two. Substituting in 𝑥 equals two gives us 𝑦 is equal to 18 multiplied by one-third squared. One-third squared is equal to one-ninth as one squared is one. And three squared is nine. 18 multiplied by one-ninth or one-ninth of 18 is two. This means that the equation does work for the third column, 𝑥 equals two, 𝑦 equals two. We could repeat this process for the last column. Substituting in 𝑥 equals three gives us an answer of 𝑦 equals two-thirds.
The correct exponential equation is 𝑦 equals 18 multiplied by one-third to the power of 𝑥.