# Write an expression in simplified radical form

Recall the three expressions in division: If we are asked to arrange the expression in descending powers, we would write. Click Here for Practice Problems Example 5 — Simplify: Step 1: Find the prime factorization of the number inside the radical and factor each variable inside the radical. Then we took the second term of the first set and multiplied it by each term of the second set, and so on. We just do not bother to write an exponent of 1. It is true, in fact, that every positive number has two square roots. Find the square roots of This can be very important in many operations. Note that when factors are grouped in parentheses, each factor is affected by the exponent. Then arrange the divisor and dividend in the following manner: Step 2: To obtain the first term of the quotient, divide the first term of the dividend by the first term of the divisor, in this case. That is the reason the x3 term was missing or not written in the original expression.

As in arithmetic, division is checked by multiplication. Note that this is an application of the distributive property. Notice, we say 5 is a square root and not the square root.

## Radical expressions and rational exponents

So we can just apply that over here: the fourth root of all of this is equal to 5 a to the 4th, b to the 12th power, all of that to the one fourth power. We just do not bother to write an exponent of 1. We now introduce a new term in our algebraic language. Then we took the second term of the first set and multiplied it by each term of the second set, and so on. Use the third law of exponents. In such an example we do not have to separate the quantities if we remember that a quantity divided by itself is equal to one. From the preceding examples we can generalize and arrive at the following law: Third Law of Exponents If a and b are positive integers and x is a nonzero real number, then If we try to use only the part of the law that states on an expression such as we would obtain At this point negative exponents have not been defined. Remember that the 1 must be written if it is the only term in the numerator. In words, "to raise a power of the base x to a power, multiply the exponents. This is very important! To divide a polynomial by a binomial use the long division algorithm. Again, each factor must be raised to the third power. Division of two numbers can be indicated by the division sign or by writing one number over the other with a bar between them.

The commutative property allows the rearrangement of order. If we sum the term a b times, we have the product of a and b.

So here we have the fourth root of 5 a to the fourth b to the twelfth power. This process is repeated until either the remainder is zero as in this example or the power of the first term of the remainder is less than the power of the first term of the divisor. Find the square roots and principal square roots of numbers that are perfect squares. Placing 2x directly in front of the parentheses means to multiply the expression in the parentheses by 2x. For the present time we are interested only in square roots of perfect square numbers.

Note in the following examples how this law is derived by using the definition of an exponent and the first law of exponents.

## How to find simplest radical form

If a polynomial has three terms it is called a trinomial. In the process of removing parentheses we have already noted that all terms in the parentheses are affected by the sign or number preceding the parentheses. Then arrange the divisor and dividend in the following manner: Step 2: To obtain the first term of the quotient, divide the first term of the dividend by the first term of the divisor, in this case. We will discuss them later. To divide a polynomial by a monomial divide each term of the polynomial by the monomial. In this case x divides into x2 x times. Multiply all numbers and variables inside the radical together.

Note that each exponent must be multiplied by 4. Again, each term in the parentheses is multiplied by 3y2 Again, each term in the parentheses is multiplied by 3y2.

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