Theory of Exponents
Review the proofs, for positive integral exponents, of:
I.
II.
III.
IV.
V.
VI.
To find the meaning of a fractional exponent.
Assume that Law I holds for all exponents.
If so,
Hence, is one of the three equal factors (hence the cube root) of In the same way,
Hence, is one of the five equal factors (hence the fifth root) of In the same way, in general,
Hence, the numerator of a fractional exponent indicates the power, the denominator indicates the root.
To find the meaning of a zero exponent.
Assume that Law II holds for all exponents.
If so, But by division,
Axiom I.
To find the meaning of a negative exponent.
Assume that Law I holds for all exponents.
If so,
Hence,
Rules:
To multiply quantities having the same base, add exponents.
To divide quantities having the same base, subtract exponents.
To raise a quantity to a power, multiply exponents.
To extract a root, divide the exponent of the power by the index of the root.
1. Find the value of
2. Find the value of
Give the value of each of the following:
3.
4. Express as some power of 7 divided by itself.
Simplify:
5. (Change to the same base first.)
6.
7.
8.
9.
10.
11.
Reference: The chapter on Theory of Exponents in any algebra.
Solve for x:
1.
2.
Factor:
3.
4.
5.
6.
7. Find the H. C. F. and L. C. M. of
8. Simplify the product of:
and (Princeton.)
9. Find the square root of:
10. Simplify
11. Find the value of
12. Express as a power of 2:
13. Simplify
14. Simplify
15. Expand writing the result with fractional exponents.
Reference: The chapter on Theory of Exponents in any algebra.