Given that four to the 𝑥 minus 28 power is equal to one over four, find the value of the cube root of 𝑥.
If we know that four to the 𝑥 minus 28 power equals one-fourth and we’re trying to solve for the cube root of 𝑥, we’ll need to first identify what the value of 𝑥 is. And to do that, we’ll need to isolate this 𝑥-value. We recognize that four to the 𝑥 minus 28 power is equal to one-fourth, which should clue us in to some exponent properties. That is, one over 𝑥 to the 𝑎 power is equal to 𝑥 to the negative 𝑎 power.
We could say that this was equal to one over four to the first power. And we can rewrite that with a negative exponent as four to the negative one power. And at this point, we have two equal bases. And if the two bases are equal, we can set these exponent values equal to one another. 𝑥 minus 28 should equal negative one. To solve for 𝑥, we add 28 to both sides of the equation, and we have 𝑥 equals 27.
And now we’re ready to take this value and plug it in to the equation we’re trying to solve. We would be solving the cube root of 27, and the cube root of 27 is three. We know that the cube root of 27 is three because three cubed is 27. Based on the information we were given, the cube root of the 𝑥-value is three.