Algebra Review

INTEGER EXPONENTS

The basic laws for working with exponents are

Example: Simplify (x2 )3 - x2 (x4 + x3 )
Solution:

Example: Eliminate negative exponents and simplify (3x8y-6)(-5x-4y6).
Solution:

Example: Eliminate negative exponents and simplify

Solution:

Example: Eliminate negative exponents and simplify

Solution:

Problems: Eliminate negative exponents and simplify.

RATIONAL EXPONENTS AND RADICALS

Recall that for a > 0 and n a positive integer, means a positive number b such that b^n = a. It follows that
Thus, radical expressions can be converted to expressions involving fractional exponents. These fractional exponents obey the
same laws as integer exponents previously
reviewed. For instance,

A word of warning! , not .

Example: Simplify by expressing radicals as non-negative rational powers and combining powers
whenever possible:

Solution:

Problems: Simplify by expressing radicals as non-negative rational powers and combining powers
whenever possible.

For typographical convenience and historical reasons, it is customary to write expressions involving both fractions and radicals
so that no fraction appears under a radical sign and no radical appears in a denominator. This "rationalizing the denominator"
is accomplished in several ways, as illustrated in the following examples.

Example:

Example:

Example:

Example:

Problems:

98. Rationalize the denominator in .
99. Rationalize the denominator in .
100. Rationalize the denominator in .
101. Rationalize the denominator in .

SOLVING RADICAL EQUATIONS

Equations involving radicals are often solved by rewriting the equation, raising both sides to some
power to remove the radical, and then solving the resulting equation. As with fractional equations, this
process may introduce extraneous roots. Potential solutions must be checked in the original equation.

Example: Solve for x:

Solution:

Squaring both sides we get 3x + 4 = 64,
so 3x = 60.
Thus, x = 20 is a possible solution.
Checking in the original equation we find . Thus x = 20 is the solution.

Example: Solve for x:

Solution:

Squaring both sides we get 5x - 1 = 9,
so 5x = 10.
Thus, x = 2 is a possible solution.
Check in the original equation and find . (Recall that ยง means nonnegative
square root!) Thus, x = 2 is not a solution. This equation has no solution.

Example: Solve for y:

Solution:

Square both sides and find

Isolate the square root term to get

and square again to get
16y2 - 48y + 36 = 36y + 108.
We find
16y2 - 84y - 72 = 0,
or 4y2 - 21y - 18 = 0,
so y = -3/4 or 6.

Checking by substituting into the original equation, we find that 6 is a solution and -3/4
is not. The only solution is y = 6.

Problems:
102. Solve for x:
103. Solve for x:
104. Solve for x:
105. Solve for x:

POWER FUNCTIONS

A function, f (x) = x^a where a is a real number is called a power function. You have already worked with many power
functions including functions like ,.

Example:

Example: Solve for x: x3/2 = 27
Solution:


Example: Solve for x:
Solution:


Problems:
106. Solve for x:
107. Solve for x:
108. Solve for x:
109. Solve for x: