## Lesson 4: Working with Radicals

In order to achieve exact answers, an answer must be left in radical form. As a result an important math skill is to know how to work with radicals.

One important skill is knowing how to place a radical in its simplest form. This involves expressing the radical as a product of two radicals and the expressing

the perfect square radical as a whole number. Let’s look at an example:

√50

=√25x2 (Note that 25 is the largest perfect square that is a factor of 50.)

=√25*√2

=5*√2

=5√2

As a result, 5√2 is the simplified form of √50. Remember to make sure one of the radicals is the greatest perfect square that is a factor of the original radicand. Adding and subtracting radicals is the same as adding or subtracting polynomials. You can only add or subtract like radicals. Like radicals are radicals with the same radicand under the √ symbols, such as √7 and 5√7. Simply add or subtract the whole number in front of the radical. you may need to fully simplify both radicals if they are not fully simplified. Remember, always work with simplified radicals. Let's look at an example where simplification is necessary:

5√8+3√18 Remember to simplify both radicals first.

=5√4*2+3√9*2

=5*2√2+3*3√2

=10√2+9√2

=19√2

Multiplying radicals is also similar to multiplying polynomials; all terms are multiplied. Whole numbers are multiplied with whole numbers and numbers under the radical sign are multiplied with numbers under the radical sign. Multiplication properties for more complicated multiplications, such as the distributive property, still hold true. With multiplication you may have to simplify your final answer. Let's look at an example:

2√3(4+5√3) Use the distributive property in this case.

=2√3(4) +(2√3)(5√3)

=8√3+10√9

=8√3+10(3)

8√3+30

Note that coefficients were multiplied with coefficients and radicals are multiplied with radicals. Afterwards, it is often possible to simply your answer. The answers you will get when working with radicals are exact; unlike rounded decimals, they are not approximations.

One important skill is knowing how to place a radical in its simplest form. This involves expressing the radical as a product of two radicals and the expressing

the perfect square radical as a whole number. Let’s look at an example:

√50

=√25x2 (Note that 25 is the largest perfect square that is a factor of 50.)

=√25*√2

=5*√2

=5√2

As a result, 5√2 is the simplified form of √50. Remember to make sure one of the radicals is the greatest perfect square that is a factor of the original radicand. Adding and subtracting radicals is the same as adding or subtracting polynomials. You can only add or subtract like radicals. Like radicals are radicals with the same radicand under the √ symbols, such as √7 and 5√7. Simply add or subtract the whole number in front of the radical. you may need to fully simplify both radicals if they are not fully simplified. Remember, always work with simplified radicals. Let's look at an example where simplification is necessary:

5√8+3√18 Remember to simplify both radicals first.

=5√4*2+3√9*2

=5*2√2+3*3√2

=10√2+9√2

=19√2

Multiplying radicals is also similar to multiplying polynomials; all terms are multiplied. Whole numbers are multiplied with whole numbers and numbers under the radical sign are multiplied with numbers under the radical sign. Multiplication properties for more complicated multiplications, such as the distributive property, still hold true. With multiplication you may have to simplify your final answer. Let's look at an example:

2√3(4+5√3) Use the distributive property in this case.

=2√3(4) +(2√3)(5√3)

=8√3+10√9

=8√3+10(3)

8√3+30

Note that coefficients were multiplied with coefficients and radicals are multiplied with radicals. Afterwards, it is often possible to simply your answer. The answers you will get when working with radicals are exact; unlike rounded decimals, they are not approximations.

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