Algebra II : Understanding Radicals

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #41 : Understanding Radicals

Simplify:  

Possible Answers:

Correct answer:

Explanation:

Multiply the integers and combine the radicals together by multiplication.

Break up square root of 800 by common factors of perfect squares.

Simplify the possible radicals.

The answer is:  

Example Question #42 : Understanding Radicals

Evaluate:  

Possible Answers:

Correct answer:

Explanation:

Multiply the integers and the value of the square roots to combine as one radical.

Simplify the radical.  Use factors of perfect squares to simplify root 300.

The answer is:  

Example Question #43 : Understanding Radicals

Simplify:  

Possible Answers:

Correct answer:

Explanation:

In order to simplify the radical, we will need to pull out common factors of possible perfect squares.

The expression becomes:  

The radical 14 does not have any common factors of perfect squares.

The answer is:  

Example Question #44 : Understanding Radicals

Simplify:  

Possible Answers:

Correct answer:

Explanation:

Simplify by factoring both radicals by perfect squares.

Replace the terms.

Combine like-terms.

The answer is:  

Example Question #45 : Understanding Radicals

Solve:  

Possible Answers:

Correct answer:

Explanation:

Multiply the integers outside of the radical.

Multiply all the values inside the radicals to combine as one radical.

Rewrite the radical using factors of perfect squares.

The answer is:  

Example Question #46 : Understanding Radicals

Simplify:  

Possible Answers:

Correct answer:

Explanation:

The values of the radicals can be combined my multiplication.

The value of  can be factored by using known perfect squares as factors.

Replace the term.

The answer is:  

Example Question #47 : Understanding Radicals

Simplify:  

Possible Answers:

Correct answer:

Explanation:

Rewrite each radical as common factors using known perfect squares.

Simplify the expression.

The answer is:  

Example Question #48 : Understanding Radicals

True or false:  is a radical expression in simplest form.

Possible Answers:

True

False

Correct answer:

True

Explanation:

A radical expression which is the th root of a constant is in simplest form if and only if, when the radicand is expressed as the product of prime factors, no factor appears  or more times. Since  is a cube, or third, root, find the prime factorization of 52, and determine whether any prime factor appears three or more times.

52 can be broken down as

and further as

No prime factor appears three or more times, so  is in simplest form.

Example Question #49 : Understanding Radicals

Try without a calculator.

Simplify:

Possible Answers:

Correct answer:

Explanation:

First, apply the Quotient of Radicals Property to split the radical into a numerator and a denominator:

Since we are dealing with cube, or third, roots, rationalize the denominator by multiplying both halves of the fraction by the least cube-root radical expression that would eliminate the radical in the denominator. To do this, note that 7 is a prime number. Therefore, to get a perfect cube, it is necessary to multiply both halves by , and apply the Product of Radicals Property. The reason for this is made more apparent below:

Example Question #50 : Understanding Radicals

Try without a calculator.

Simplify:

Possible Answers:

Correct answer:

Explanation:

First, apply the Quotient of Radicals Property to split the radical into a numerator and a denominator:

8 is a perfect cube -  - so the denominator can be simplified:

To simplify the numerator, find the prime factorization of its radicand, 250, and look for any prime factors that appear three times:

5 appears three times, so the numerator can be simplified by way of the Product of Radicals Property:

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