When multiplying square roots, you are allowed to multiply the numbers inside the square root. Then simplify if necessary. To multiply square roots, we multiply the numbers inside the radical and we can simplify them if possible. If you've found an issue with this question, please let us know.
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Email address: Your name:. Possible Answers:. Correct answer:. Explanation : When multiplying square roots, you are allowed to multiply the numbers inside the square root. Report an Error. Explanation : To simplify the problem, just distribute the radical to each term in the parentheses. When we are given expressions that involve radicals in the denominator, it makes it easier to evaluate the expression if we rewrite it in a way that the radical is no longer in the denominator.
This process is called rationalizing the denominator. Before we begin, remember that whatever we do to one side of an algebraic equation, we must also do to the other side. This same principle can be applied to fractions: whatever we do to the numerator, we must also do to the denominator, and vice versa.
Recall that a radical multiplied by itself equals its radicand, or the value under the radical sign. A radical expression represents the root of a given quantity.
There is no real value such that when multiplied by itself it results in a negative value. That is where imaginary numbers come in. When the radicand the value under the radical sign is negative, the root of that value is said to be an imaginary number.
Here are some examples:. Privacy Policy. Skip to main content. Numbers and Operations. Search for:. Learning Objectives Describe the root of a number in terms of exponentiation. Key Takeaways Key Points Roots are the inverse operation of exponentiation. The square root of a value is the number that when squared results in the initial value. The two radicals have different roots, so you cannot multiply the product of the radicands and put it under the same radical sign.
So, this problem and answer pair is incorrect. Dividing Radical Expressions. You can use the same ideas to help you figure out how to simplify and divide radical expressions. Recall that the Product Raised to a Power Rule states that. Well, what if you are dealing with a quotient instead of a product? There is a rule for that, too. The Quotient Raised to a Power Rule states that. Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well: , so.
A Quotient Raised to a Power Rule. As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like. Use the rule to create two radicals; one in the numerator and one in the denominator. Simplify each radical. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors. Identify and pull out perfect squares. Rewrite using the Quotient Raised to a Power Rule.
Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. Identify and pull out perfect cubes. You can simplify this expression even further by looking for common factors in the numerator and denominator.
Rewrite the numerator as a product of factors. Identify factors of 1, and simplify. That was a lot of effort, but you were able to simplify using the Quotient Raised to a Power Rule. What if you found the quotient of this expression by dividing within the radical first, and then took the cube root of the quotient?
Since both radicals are cube roots, you can use the rule to create a single rational expression underneath the radical. Within the radical, divide by Identify perfect cubes and pull them out. As with multiplication, the main idea here is that sometimes it makes sense to divide and then simplify, and other times it makes sense to simplify and then divide.
Whichever order you choose, though, you should arrive at the same final expression. Notice that the process for dividing these is the same as it is for dividing integers. Use the Quotient Raised to a Power Rule to rewrite this expression.
Simplify by identifying similar factors in the numerator and denominator and then identifying factors of 1. Identify perfect cubes and pull them out of the radical. As you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. For example, while you can think of as equivalent to since both the numerator and the denominator are square roots, notice that you cannot express as.
In this second case, the numerator is a square root and the denominator is a fourth root. Divide and simplify. Using what you know about quotients, you can rewrite the expression as , simplify it to , and then pull out perfect squares. The simplified form is.
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