29 RATIONAL EXPONENTSExponential form vs. radical form BY THE CUBE ROOT of a, we mean that number whose third power is a. Thus the cube root of 8 is 2, because 23 = 8. The cube root of −8 is −2 because (−2)3 = −8.
In general,
Equivalently, Read For example,
If the index is omitted as in
We see that, if the index is odd, then the radicand may be negative. But if the index is even, the radicand may not be negative. There is no such real number as Problem 1. Evaluate each the following -- if it is real.
Fractional exponent We have seen that to square a power, double the exponent. (a4)2 = a8. Conversely, then, the square root of a power will be half the exponent. The square root of a8 is a4; of a10 is a5; of a12 is a6. This wil[l hold for all powers. The square root of a3 is a In other words,
Similarly, since the cube of a power will be the exponent multiplied by 3—the cube of an is a3n—the cube root of a power will be the exponent divided by 3. The cube root of a6 is a2; of a2 is a a Here is the formal rule:
The denominator of a fractional exponent
8
Problem 3. Evaluate the following.
Problem 4. Express each radical in exponential form
a a That is, To evaluate a fracitional exponent, it is more efficient to take the root first; for we will then take the power of a smaller number. For example, 8 8 Again: The denominator of a fractional exponent In general, Problem 5. Evaluate the following.
Problem 6. Express each radical in exponential form.
Negative exponent A number with a negative exponent is defined to be the reciprocal of that number with a positive exponent.
a−v is the reciprocal of av. Therefore,
Problem 7. Express each of the following with a negative exponent.
Problem 8. Express in radical form.
In the Lesson on exponents, we saw that −24 is a negative number. It is the negative of 24. A minus sign signifies the negative of the number that follows. And the number that follows the minus sign here, −24, is 24. Similarly, then, −8 −8 (−8) (−8) Problem 9. Evaluate the following.
It is the reciprocal of 16/25 -- with a positive exponent. The rules of exponents An exponent may now be any rational number. Rational exponents u, v will obey the usual rules.
Example 3. Rewrite in exponential form, and apply the rules.
See Skill in Arithmetic, Adding and Subtracting Fractions. Problem 11. Apply the rules of exponents.
Problem 12. Express each radical in exponential form, and apply the rules of exponents.
We can now understand that the rules for radicals -- specifically, -- are rules of exponents. As such, they apply only to factors.
Problem 13. Prove:
To solve an equation that looks like this:
For, x Problem 14. Solve for x.
![]() Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: teacher@themathpage.com |