Radicals - Algebra 2

Multiply by on both sides.

Divide by 4 on both sides.

Square both sides to eliminate the radical.

Divide by three on both sides. This is similar to multiplying both sides by one third.

The answer is:

Multiply both sides by the denominator on the left side.

The equation becomes:

Square both sides to eliminate the radical.

Divide both sides by eight.

Notice that the negative radical solution will not satisfy the original equation.

Rationalize the positive term.

The answer is:

Raise both sides to the power of four.

Simplify both sides.

Subtract 1 from both sides.

Divide by 32 on both sides.

Reduce this fraction.

The answer is:

In order to eliminate the radical on both sides, we will need to cube both sides.

Simplify both sides.

Add two on both sides.

Divide by eight on both sides.

Reduce the fractions on both sides.

The answer is:

Square both sides to eliminate the radical.

Subtract three on both sides.

The equation becomes:

Divide by negative five on both sides.

The answer is:

To add or subtract radicals, they must be the same root and have the same number under the radical before combining them. Look for perfect squares that divide into the number under the radicals because those can be simplified.

Take the square roots of each of the perfect squares in these radicals and bring it out of the radical. It will multiply to any coefficient in front of that radical

Remember, only radicals with the same number can be combined

This is the final answer.

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:

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:

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: