Write an exponential equation in the form 𝑦 is equal to 𝑏 to the power of 𝑥 for the numbers in the table.
We are given four sets of values. When 𝑥 equals zero, 𝑦 is equal to one. When 𝑥 equals one, 𝑦 is two-fifths. When 𝑥 equals two, 𝑦 is four over 25. And when 𝑥 equals three, 𝑦 is equal to eight over 125. We can calculate the value of 𝑏 in the equation by substituting in our values. We begin with 𝑥 equals zero and 𝑦 equals one. This gives us one is equal to 𝑏 to the power of zero. However, anything to the power zero is equal to one, so this does not help us calculate the value of 𝑏.
Moving on to the second pair of values, we have two-fifths is equal to 𝑏 to the power of one. Anything to the power of one is itself. Therefore, 𝑏 is equal to two-fifths. Substituting this back into our equation gives us 𝑦 is equal to two-fifths to the power of 𝑥. We can then check this answer by substituting in our third and fourth pair of values.
When 𝑥 equals two and 𝑦 is four twenty-fifths or four over 25, we have four over 25 is equal to two-fifths squared. When squaring a fraction, we can square the numerator and denominator separately. Two squared is equal to four, and five squared is 25. This means that the formula is correct for this pair of values. Finally, we have eight over 125 is equal to two-fifths cubed. Once again, we can split the numerator and denominator. So we have two cubed over five cubed. Two cubed is equal to eight, and five cubed is 125. Therefore, this pair of values is also correct. The exponential equation for the numbers in the table is 𝑦 is equal to two-fifths to the power of 𝑥.