Simplify radical expressions
When we write x, the exponent is assumed: x = x1. This law applies only when this condition is met.Īn exponent of 1 is not usually written. Note in the above law that the base is the same in both factors. To multiply factors having the same base add the exponents.įor any rule, law, or formula we must always be very careful to meet the conditions required before attempting to apply it. These laws are derived directly from the definitions.įirst Law of Exponents If a and b are positive integers and x is a real number, then Now that we have reviewed these definitions we wish to establish the very important laws of exponents. Upon completing this section you should be able to correctly apply the first law of exponents. MULTIPLICATION LAW OF EXPONENTS OBJECTIVES We just do not bother to write an exponent of 1. It is also understood that a written numeral such as 3 has an exponent of 1. This can be very important in many operations. When we write a literal number such as x, it will be understood that the coefficient is one and the exponent is one. Many students make the error of multiplying the base by the exponent.For example, they will say 3 4 = 12 instead of the correct answer, Note that only the base is affected by the exponent. Unless parentheses are used, the exponent only affects the factor directly preceding it. From using parentheses as grouping symbols we see thatĢx 3 means 2(x)(x)(x), whereas (2x) 3 means (2x)(2x)(2x) or 8x 3. Note the difference between 2x 3 and (2x) 3. An exponent is usually written as a smaller (in size) numeral slightly above and to the right of the factor affected by the exponent.Īn exponent is sometimes referred to as a "power." For example, 5 3 could be referred to as "five to the third power." Make sure you understand the definitions.Īn exponent is a numeral used to indicate how many times a factor is to be used in a product. When naming terms or factors, it is necessary to regard the entire expression.įrom now on through all algebra you will be using the words term and factor. Rules that apply to terms will not, in general, apply to factors. It is very important to be able to distinguish between terms and factors. When an algebraic expression is composed of parts to be multiplied, these parts are called the factors of the expression. In 2x + 5y - 3 the terms are 2x, 5y, and -3. When an algebraic expression is composed of parts connected by + or - signs, these parts, along with their signs, are called the terms of the expression. Since these definitions take on new importance in this chapter, we will repeat them.
Go to the next section to see examples with cube roots.In section 3 of chapter 1 there are several very important definitions, which we have used many times. Next, clean up the problem to get the final answer. Place a factor on the outside of the radical for each group. The index is 2, so we will group in pairs. Now, we will write the prime factors under a radical. This is a factor tree for the radicand 72. Anything ungrouped will remain under the radical, like so. This means we group the factors in groups of two, like so.įor each group, a factor comes out. So, we will write these in place of the 12 under the radical. However, 4 is not prime (called composite) and it can be factored as follows. 3 is prime because it has no other smaller factors other than 1. Our only other option is to simplify the radical using the steps outlined within our graphic organizer in the last section, Steps for Simplifying Radical Expressions. If we were to use a calculator to find the solution, it would provide us with a decimal solution somewhere between 3 and 4.
This problem does not have a perfect answer.
#Simplify radical expressions how to
The best way to learn how to simplify an expression is to examine an example.