Algebra II : Understanding Radicals

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #21 : Understanding Radicals

Evaluate:  

Possible Answers:

Correct answer:

Explanation:

Evaluate each square root.  The square root of a number evaluates into a number which multiplies by itself to achieve the number in the square root.

Substitute the terms back into the expression.

The answer is:  

Example Question #22 : Understanding Radicals

Solve:  

Possible Answers:

Correct answer:

Explanation:

Solve each radical.  The square root determines a number that multiples by itself to equal the number inside the square root.

Rewrite the expression.

The answer is:  

Example Question #21 : Radicals

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

Possible Answers:

False

True

Correct answer:

False

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 square, or second, root, find the prime factorization of 52, and determine whether any prime factor appears two or more times.

52 can be broken down as

and further as

The factor 2 appears twice, so  is not in simplest form.

Example Question #21 : Understanding Radicals

Possible Answers:

Correct answer:

Explanation:

To solve this, remember that when multiplying variables, exponents are added.  When raising a power to a power, exponents are multiplied.  Thus:

Example Question #2 : Non Square Radicals

Simplify by rationalizing the denominator:

Possible Answers:

Correct answer:

Explanation:

Since , we can multiply 18 by  to yield the lowest possible perfect cube:

Therefore, to rationalize the denominator, we multiply both nuerator and denominator by  as follows:

Example Question #3 : Non Square Radicals

Simplify: 

Possible Answers:

Correct answer:

Explanation:

Begin by getting a prime factor form of the contents of your root.

Applying some exponent rules makes this even faster:

Put this back into your problem:

Returning to your radical, this gives us:

Now, we can factor out  sets of  and  set of .  This gives us:

Example Question #4 : Non Square Radicals

Simplify:

Possible Answers:

Correct answer:

Explanation:

Begin by factoring the contents of the radical:

This gives you:

You can take out  group of .  That gives you:

Using fractional exponents, we can rewrite this:

Thus, we can reduce it to:

Or:

Example Question #22 : Understanding Radicals

Simplify: 

Possible Answers:

Correct answer:

Explanation:

To simplify , find the common factors of both radicals.

Sum the two radicals.

The answer is:  

Example Question #6 : Non Square Radicals

Simplify:

Possible Answers:

Correct answer:

Explanation:

To take the cube root of the term on the inside of the radical, it is best to start by factoring the inside:

Now, we can identify three terms on the inside that are cubes:

We simply take the cube root of these terms and bring them outside of the radical, leaving what cannot be cubed on the inside of the radical.

Rewritten, this becomes

Example Question #7 : Non Square Radicals

Simplify the radical:  

Possible Answers:

Correct answer:

Explanation:

Simplify both radicals by rewriting each of them using common factors.

Multiply the two radicals.

The answer is:  

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