A Few Worked Problems

Here are a few more worked problems. Study them to acquaint yourself with the use of different operations you can perform with exponents.

• This problem combines a cube root with the multiplicative inverse of a value:

  

  In this instance, the negative exponent (–1/3) calls for the multiplicative inverse of the cube root of 8. Having expressed the problem in this form, you can then extract the root and arrive at 1/2.

• This problem involves working with both a negative exponent and the need to simplify an expression that involves a root and a power:

  

  With this problem, you begin by dealing with the negative exponent. You change the exponent into radical notation and then work on simplifying the radical. There are a few approaches to this. One involves emphasizing that You can then raise 2 to the third power, which results in 8. An alternative approach is to write the denominator as

  

• This problem involves a situation in which you must carry out a multiplication operation in the denominator of the fraction.

  

  To carry out the multiplication, since you are working with the same base (4), you can add the exponents (2/3 + 5/3) of the two numbers in the denominators. Then, to carry out the division, you can subtract the exponent of the denominator from the exponent of the numerator. The result is 0, and any number raised to the power of 0 is equal to 1.

• This problem has a denominator that includes several radical expressions:

  [View full size image]
  

  Resolution of the problem involves first moving the numerator to the denominator, and then adding the two exponents of 25. When you add 1/3 and 2/3, the result is 1, and any number raised to the power of 1 is the number itself. When you express the fractional exponent of 8 in a radical form, you see clearly that you are looking for the cube root of 8, which is 2.