Algebra 1 End-of-Course Review
Module Ten: Radical Expressions
In this last module, you will review algebraic ratios and proportions, as well as radical expressions.
Simplifying Algebraic Ratios
To simplify a ratio of monomials you must cancel all common factors that appear in the numerator and denominator. Recall, instead of writing each power in expanded form to cancel common factors, use the division property of exponents for a faster way of dividing powers of the same base.
So how would you go about simplifying a ratio of polynomials, such as 
?
When simplifying ratios involving polynomials, keep in mind that only factors that are being multiplied can be cancelled out. Terms that are being added cannot be cancelled out. For example: You cannot cancel the x’s in the ratio 
without changing the value of the expression.
Check out some more examples of simplifying a ratio of polynomials.
Solving Algebraic Proportions
Follow these steps to solve algebraic proportions:
Step 1: Simplify if possible.
Step 2: Cross-multiply.
Step 3: Solve for x.
Step 4: Check your work.
Check out a couple of examples using these four steps to solve some algebraic proportions.
Sometimes the products of cross-multiplication can only be simplified by using quadratic equations. When that is the case, all that changes is how you solve for x.
Simplifying Radical Expressions
When the radicand of a radical expression is a perfect square, you can simplify the expression by figuring out what number times itself results in that square as the product. When the radicand is not a perfect square, you can still simplify the expression using the greatest perfect squares method or the
prime factorization method.
Simplifying When the Radicand Includes Variables
The square root of a variable with an even exponent simplifies to the same variable, but with an exponent that is half the power of the variable in the radicand.
For example: 
= x3.
If the exponent is odd, subtract 1 from the exponent so that your exponent is even. Then rewrite the radicand as factors in which one factor is the even exponent and the other is the variable (the 1 which you subtracted). Find the square root of the even exponent by dividing it by 2.
Whenever you see both a coefficient and a variable inside a square root, simplify each one separately, and then combine them in the end.
Take a look at some examples of simplifying radicals with coefficients and variables.
Adding and Subtracting Radical Expressions
Adding and subtracting radical expressions is the same as combining like terms except that the terms contain radical parts instead of variable parts.
If you are uncertain whether two terms including variables could be considered like terms, try simplifying each one first. Then compare the simplified terms. The terms under the radical sign are like terms only if they are exactly the same (coefficients and variables).
Take a look at an example of simplifying a radical expression with like radicands.
What about when the radical expression contains a variable? Good news...You can add and subtract the same way that you did for numeric radical expressions! Check out an example.
By simplifying each term of the expression separately, you can even perform addition and subtraction on what may appear to be unlike terms involving radicals.
Example
Find the exact value of 3
+ 4
– 
Take a look at one more example of simplifying a radical expression using addition and subtraction.
Multiplication with Radical Expressions
As long as you are taking the square root of two positive numbers, you can say 
• 
= 
. This is true even when the numbers don’t multiply to perfect squares. For example, 
• 
= 
.
When multiplying radicals that include numbers outside the square root, there is one important rule to follow:
In other words, multiply values that are outside the radical and values that are inside the radical separately.
Sometimes, a factor in a multiplication problem can be simplified before multiplying. When this happens, all you have to do is include one more step in which you simplify first. Check out the example below.
Multiply: 
• 9
Distributing Radical Expressions
Sometimes multiplication is only part of a radical expression. Remember that the distributive property requires both addition and multiplication. When you have to add and multiply radicals, just stick to the rules for each one separately!
Check out an example where the square root of two is distributed to a radical expression.
Basic Division with Radical Expressions
The quotient property of radicals states that the square root of a quotient is equal to the quotient of the square roots. In other words:
= 
, where b ≠ 0.
Example
Find the value of 
.
- Method 1
- Method 2
Work through an example that combines variables and coefficients in the radicand.
Radicals in the Denominator
The solution to a radical expression must always be written in simplest form. A radical expression is in simplest form when
- the radicand contains no perfect square factors
- the radicand contains no fractions
- the denominator contains no radicals
Recall that when the square root of any number is multiplied by itself, or squared, the radical is eliminated. To eliminate a square root from the denominator of an expression, you can multiply the denominator by the radical it contains. To maintain equality, you must also multiply the numerator by the same radical. This method is called rationalizing the denominator. Recall that any number over itself is equal to 1. So in essence, you are multiplying by 1.
Example
Simplify 
.
Check out one more example of rationalizing the denominator.
