Understanding Radicals - Algebra II

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent as shown below:

In this case, we are done because there are no further simplification steps.

In order to solve this expression, we will need to rewrite each square root using one half as the exponent. This will allow us to solve for the value of the inner terms.

We can then rewrite the radical using factors of perfect numbers raised to the fourth power, such as:

The answer is:

Evaluate each square root. The square root of the number is equal to a number multiplied by itself.

The answer is:

The first term is already simplified. The second and third term will need to be simplified.

Write the common factors of the second radical and simplify.

Repeat the process for the third term.

Rewrite the expression.

Combine like-terms.

The answer is:

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

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:

To rationalize a denominator, multiply all terms by the conjugate. In this case, the denominator is , so its conjugate will be .

So we multiply: .

After simplifying, we get .

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:

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:

To simplify , find the common factors of both radicals.

Sum the two radicals.

The answer is:

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

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

Multiply the two radicals.

The answer is:

In order to simplify this radical, rewrite the radical using common factors.

Simplify the square roots.

Multiply the terms inside the radical.

The answer is:

Break down the two radicals by their factors.

A square root of a number that is multiplied by itself is equal to the number inside the radical.

Simplify the terms in the parentheses.

The answer is:

Simplify the first term by using common factors of a perfect square.

Simplify the second term also by common factors.

Combine the terms.

The coefficients cannot be combined since these are unlike terms.

The answer is:

To simplify this, multiply the top and bottom by the denominator.

Reduce the fraction.

The answer is:

To simplify radicals, we need to find perfect squares to factor out. In this case, it's .

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To simplify radicals, we need to find perfect squares to factor out. In this case, it's .

To simplify radicals, we need to find perfect squares to factor out. In this case, it's .

To simplify radicals, we need to find perfect squares to factor out. In this case, it's .