Simplify the expression four 𝑥 to the power of one-quarter divided by two 𝑥 to the power of negative two-thirds.
In order to answer this question, we’ll begin by recalling one of our laws of exponents or indices. This states that 𝑎 to the power of 𝑏 divided by 𝑎 to the power of 𝑐 is equal to 𝑎 to the power of 𝑏 minus 𝑐. Before using this rule, we will divide the integers. Four divided by two is two. Using the rule, we can rewrite 𝑥 to the power of one-quarter divided by 𝑥 to the power of negative two-thirds as 𝑥 to the power of one-quarter minus negative two-thirds.
This is the same as adding one-quarter and two-thirds. In order to add two fractions, we need a common denominator. As the lowest common multiple of four and three is 12, we will have a common denominator of 12. As we have multiplied the denominator of the first fraction by three, we need to do the same to the numerator. And we need to multiply the numerator of the second fraction by four, giving us three twelfths plus eight twelfths which is equal to eleven twelfths. The exponent one-quarter minus negative two-thirds simplifies to eleven twelfths.
We can therefore conclude that four 𝑥 to the power of one-quarter divided by two 𝑥 to the power of negative two-thirds is equal to two 𝑥 to the power of eleven twelfths.