A radical expression is said to be in standard form if the following conditions hold:. The exponent of each factor of the radicand is a natural number less than the radical index. By simplifying a radical expression, we mean putting the radical expression in standard form. When the exponents of some factors of the radicand are greater than the radical index, but not an integral multiple of it, write each of these factors as a product of two factors one factor with an exponent that is an integral multiple of the radical index, and the other factor with an exponent that is less than the radical index.
Write the factors with exponents that are integral multiples of the index under one radical, thus obtaining a perfect root. The cases when there are fractions in the radicand and radicals in the denominator of a fraction will be discussed later. The redical expressions 3root 2 and 5root 2 are similar. The redical expressions root 24 and root 54 can be shown to be similar. The redical expressions root root 18 and root 27 are not similar.
In the next example, we practice writing radicals with rational exponents where the numerator is not equal to one. In our last example we will rewrite expressions with rational exponents as radicals. This practice will help us when we simplify more complicated radical expressions, and as we learn how to solve radical equations.
Typically it is easier to simplify when we use rational exponents, but this exercise is intended to help you understand how the numerator and denominator of the exponent are the exponent of a radicand and index of a radical. In the following video we show more examples of writing radical expressions with rational exponents and expressions with rational exponents as radical expressions. We will use this notation later, so come back for practice if you forget how to write a radical with a rational exponent.
To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the Product Raised to a Power Rule from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power.
The exponent is distributed in the same way. Look at that—you can think of any number underneath a radical as the product of separate factors , each underneath its own radical. This rule is important because it helps you think of one radical as the product of multiple radicals. Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical.
Rearrange factors so the integer appears before the radical, and then multiply. This is done so that it is clear that only the 7 is under the radical, not the 3. The following video shows more examples of how to simplify square roots that do not have perfect square radicands.
Before we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand. This looks like it should be equal to x , right? Where are they equal? Where are they not equal?
We will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.
Take the square root of each radical. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd — including 1 — add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples. In the following video you will see more examples of how to simplify radical expressions with variables.
We will show another example where the simplified expression contains variables with both odd and even powers. Because x has an odd power, we will add the absolute value for our final solution.
In our next example we will start with an expression written with a rational exponent. You will see that you can use a similar process — factoring and sorting terms into squares — to simplify this expression.
We can use the same techniques we have used for simplifying square roots to simplify higher order roots. When simplifying radical expressions, look for factors with powers that match the index.
Example 3: Simplify: 18 x 3 y 4. Solution: Begin by determining the square factors of 18, x 3 , and y 4. Make these substitutions and then apply the product rule for radicals and simplify. Example 4: Simplify: 4 a 5 b 6. Solution: Begin by determining the square factors of 4, a 5 , and b 6. Example 5: Simplify: 80 x 5 y 7 3. Solution: Begin by determining the cubic factors of 80, x 5 , and y 7. Example 6: Simplify 9 x 6 y 3 z 9 3. It will be left as the only remaining radicand because all of the other factors are cubes, as illustrated below:.
Replace the variables with these equivalents, apply the product and quotient rule for radicals, and then simplify. Example 7: Simplify: 81 a 4 b 5 4. Solution: Determine all factors that can be written as perfect powers of 4.
Hence the factor b will be left inside the radical. Solution: Notice that the variable factor x cannot be written as a power of 5 and thus will be left inside the radical. Try this! Simplify: x 6 y 7 z Assume all variables are positive. To easily simplify an n th root, we can divide the powers by the index. If the index does not divide into the power evenly, then we can use the quotient and remainder to simplify. For example,. The quotient is the exponent of the factor outside of the radical, and the remainder is the exponent of the factor left inside the radical.
We next review the distance formula. The distance, d , between them is given by the following formula:. Recall that this formula was derived from the Pythagorean theorem. Solution: Use the distance formula with the following points. It is a good practice to include the formula in its general form before substituting values for the variables; this improves readability and reduces the probability of making errors.
Answer: 6 2 units. Example The period, T , of a pendulum in seconds is given by the formula.
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