In Algebra help , An expression is all possible combinations of numbers, variables and operators. If any of the terms in an expression is under a radical sign, then the expressions is called as a radical expressions.
Radical Expressions – Basic Concepts
The symbol ‘?’ is called radical symbol. The actual number and the its root number are inserted as shown.
For example, the cube root of 8 is denoted as $\sqrt[3]{8}$
In general nth root of a number a is written in radical form as $\sqrt[n]{a}$
The number a is called as radicand.
In case of square roots, the root number 2 is not inserted and hence the mere symbol ‘?’ always means a square root.
Radical Expressions and Irrationals
Before going into the details of radical expressions, let us review the concept of irrational numbers.
An irrational number is the one which can not be expressed in the form of p/q. Roots of most of the numbers are decimal numbers with non terminating terms without a pattern. All such numbers are irrational numbers and can be written only in radical form.
Therefore, all irrational numbers can be written conveniently in radical form.
But the converse is not true. That is all numbers written in radical form need not be irrationals.
Consider the number $\sqrt{16}$. It is a radical but it is not irrational as $\sqrt{16}$= 4 and 4 can be written as 4/1.
Types of Radical Expressions
As defined earlier a radical expression contains at least one radical term.
Let us say, (x + ?y) is a simple radical expression.
The expression can be evaluated by assigning the given values for x and y.
For example if x = 1 and y = 4, the value of the expression is, 1 + ?4
By taking the positive square root (principal root) of 4 , it is simplified to 1 + 2 = 3.
Suppose the value of b is changed to 2, what happens?
The expression is evaluated to the extent, 1 + ?2 and can not be simplified further because ?2 is an irrational number. This expression is called as an expression in surd form and the irrational number ?2 is called as a surd.
A surd is defined as a radical which does not have clear number of roots. In other words surds are all irrational numbers. While all radical numbers need not be irrationals, all surds are irrationals.
Simplifying Radical Expressions
This following algebra answers gives a step by step procedure to solve the given questions
Let us again consider the simple radical expression, (x + ?y) and simplifying radical expression is as follows.
Let the value of x be 1 as constant and assign different values for y.
Assume the value assigned to y is 20.
The value of the expression is 1 + ?20.
This form of expression is called as the radical expression with a total surd as the second term is purely under a radical and which does not have clear roots.
However, ?20 may not have clear roots. But ?20 can be written as $\sqrt{(4)(5)}$. Since one of its factors 4 has a positive root as 2, ?20 can be rewritten as 2?5 which is named as a mixed surd.
Thus the given expression can be evaluated as, 1 + 2?5
This form of radical expression is called radical expression with a mixed surd.
The mixed surds with same radicand are called like surds and with different radicands are called unlike surds.
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