Write an exponential equation in the form 𝑦 equals 𝑏 to the power of 𝑥 for the numbers in the table. 𝑥 equals zero, 𝑦 equals one. 𝑥 equals one, 𝑦 equals five. 𝑥 equals two, 𝑦 equals 25. And 𝑥 equals three, 𝑦 equals 125.
There are lots of ways of approaching this question. One way would be to substitute the values of 𝑥 and 𝑦 into the equation 𝑦 equals 𝑏 to the power of 𝑥. Let’s consider the first column when 𝑥 equals zero, 𝑦 equals one. Substituting in these values gives us one is equal to 𝑏 to the power of zero. We know from our laws of exponents or indices that anything to the power of zero is equal to one. Therefore, this doesn’t help us work out the value of 𝑏.
In the next column, we’re told that 𝑥 is equal to one and 𝑦 is equal to five. This gives us the equation five is equal to 𝑏 to the power of one. Once again, from our laws of exponents, we know that anything to the power of one is equal to itself. We can therefore say that if 𝑏 to the power of one is equal to five, 𝑏 must be equal to five. As we have now calculated the value of 𝑏, we can rewrite the exponential equation as 𝑦 equals five to the power of 𝑥. Whilst this appears to be the correct answer, it is worth substituting in our values in column three and four to check that we are correct.
Substituting in 𝑥 equals two gives us 𝑦 is equal to five squared. Squaring a number is the same as multiplying it by itself. And five multiplied by five is 25. This means that the numbers 𝑥 equals two and 𝑦 equals 25 do fit the equation. Substituting in 𝑥 equals three gives us 𝑦 is equal to five cubed or five to the power of three. This is the same as five multiplied by five multiplied by five. Five cubed is therefore equal to 125. So this pair of numbers also fits the equation.
We can therefore conclude that the exponential equation 𝑦 equals five to the power of 𝑥 is correct.