How to : How to Simplify Radical Expressions

Method 1

Method 1 of 5:

Simplifying the Square Root of an Integer

1
Factor the number under the square root. Ignore the square root for now and just look at the number underneath it. Factor that number by writing it as the product of two smaller numbers. (If the factors aren’t obvious, just see if it divides evenly by 2. If not, try again with 3, then 4, and so on, until you find a factor that works.)^{[1]

X

Research source

Example: Simplify ${displaystyle {sqrt {45}}}$ .
The first step is finding some factors of 45. You can’t divide 45 by 2, so try dividing it by 3 instead: ${displaystyle 45div 3=15}$ , so ${displaystyle 45=3times 15}$ .
${displaystyle {sqrt {45}}={sqrt {3times 15}}}$}
2
Keep going until the number is factored completely. Remember, any number can be factored down into prime numbers (like 2, 3, 5, and 7). Keep breaking down the factors until there are no more factors to find.^{[2]

X

Research source

Now we have ${displaystyle {sqrt {3times 15}}}$ , but we can factor 15 again into ${displaystyle 3times 5}$ .
${displaystyle {sqrt {45}}={sqrt {3times 3times 5}}}$}
3
Rewrite pairs of the same number as powers of 2. If the same factor shows up more than once, rewrite it as an exponent. (Keep everything underneath the square root.)^{[3]

X

Research source

In ${displaystyle {sqrt {3times 3times 5}}}$ , the number 3 shows up twice. Since ${displaystyle 3times 3=3^{2}}$ , we can rewrite the whole expression as ${displaystyle {sqrt {3^{2}times 5}}}$ .}
4
Take any numbers raised to the power of 2 outside the square root. Roots and exponents are opposite, so they cancel each other out. If any factors are raised to the power of 2, move that factor in front of the square root (and get rid of the exponent).^{[4]

X

Research source

${displaystyle {sqrt {3^{2}times 5}}={sqrt {3^{2}}}{sqrt {5}}}$ (As long as everything underneath the root is one multiplication problem, you can always rewrite the expression like this, with a root over each product.)
${displaystyle {sqrt {3^{2}}}{sqrt {5}}=3{sqrt {5}}}$
Since there are no other exponents left under the square root, you’re all done!}
5
Simplify the result so there is no multiplication left. In more difficult problems, you might end up with multiple numbers in front of the square root, or underneath it. Solve these multiplication problems to simplify the answer.
- Example: Simplify ${displaystyle {sqrt {360}}}$ .
- This takes a lot of factoring to break down: ${displaystyle {sqrt {360}}={sqrt {40times 9}}={sqrt {(5times 8)times (3times 3)}}={sqrt {5times 2times 2times 2times 3times 3}}}$
- Rewrite pairs of numbers using exponents: ${displaystyle {sqrt {5times 2^{2}times 2times 3^{2}}}}$ .
- Bring the 2 and 3 outside the square root: ${displaystyle 2times 3times {sqrt {5times 2}}}$
- Simplify the numbers in front of the square root: ${displaystyle 6{sqrt {5times 2}}}$
- To get the final answer, simplify the numbers under the square root: ${displaystyle 6{sqrt {10}}}$

Method 2

Method 2 of 5:

Simplifying Cube Roots and Higher Roots

1
Find the prime factors of the number under the root. Just like square roots, the first step to simplifying a cube root ( ${displaystyle {sqrt[{3}]{}}}$ ${displaystyle {sqrt[{3}]{}}}$ ), a fourth root ( ${displaystyle {sqrt[{4}]{}}}$ ${displaystyle {sqrt[{4}]{}}}$ ), or any higher root is to factor the number under the root.^{[5]

X

Research source

Example: Simplify ${displaystyle {sqrt[{3}]{81}}}$ (the cube root of 81).

${displaystyle 81=3times 3times 3times 3}$ , so ${displaystyle {sqrt[{3}]{81}}={sqrt[{3}]{3times 3times 3times 3}}}$}
2
Rewrite groups of the same factors in exponent form. If the same prime factor shows up more than once, rewrite them as an exponent.^{[6]

X

Research source

${displaystyle {sqrt[{3}]{3times 3times 3times 3}}={sqrt[{3}]{3^{4}}}}$}
3
Simplify the root of exponents wherever possible. Just as a square root cancels out a square, higher roots cancel out matching exponents (for instance, ${displaystyle {sqrt[{3}]{}}}$ ${displaystyle {sqrt[{3}]{}}}$ and ${displaystyle ^{3}}$ ${displaystyle ^{3}}$ ). Check out this example to see how this works:^{[7]

X

Research source

${displaystyle {sqrt[{3}]{3^{4}}}={sqrt[{3}]{3^{3}times 3}}={sqrt[{3}]{3^{3}}}{sqrt[{3}]{3}}}$
Since the root and exponent match in ${displaystyle {sqrt[{3}]{3^{3}}}}$ , they cancel out, leaving only the base number, ${displaystyle 3}$ .
Plug that into the whole expression to get ${displaystyle 3{sqrt[{3}]{3}}}$ . Since there are no more exponents left that can cancel out, this is the simplified form.}
4
Simplify any multiplication and exponents. You’ll often end up with exponents that don’t cancel out, or with more than one number multiplied together. Solve for these so you end up with one number outside the radical, and one number inside it.
- Example: Simplify ${displaystyle {sqrt[{5}]{2^{7}times 3^{5}times 7}}}$ .
- This is already factored into prime numbers, so we can skip that step. Let’s rewrite this as ${displaystyle {sqrt[{5}]{2^{7}}}{sqrt[{5}]{3^{5}}}{sqrt[{5}]{7}}}$ .
- ${displaystyle {sqrt[{5}]{2^{7}}}={sqrt[{5}]{2^{5}times 2^{2}}}}$ , by the rules of exponents. The root and exponent cancel out in the first term, leaving ${displaystyle 2{sqrt[{5}]{2^{2}}}}$
- In ${displaystyle {sqrt[{5}]{3^{5}}}}$ , the root and exponent cancel out to make ${displaystyle 3}$ .
- Since ${displaystyle {sqrt[{5}]{7}}}$ has no exponents that cancel out, this can’t be simplified.
- Plug your simplified terms back into the whole expression: ${displaystyle 2{sqrt[{5}]{2^{2}}}times 3times {sqrt[{5}]{7}}}$
- Combine like terms: ${displaystyle (2times 3){sqrt[{5}]{2^{2}times 7}}}$
- Calculate multiplication and exponents: ${displaystyle 6{sqrt[{5}]{28}}}$

Method 3

Method 3 of 5:

Simplifying Fractions inside Roots

1
Simplify the fraction. What if a whole fraction is underneath a root? One way to solve problems like this is to ignore the radical expression at first. Simplify the fraction as much as you can, then see if the root lets you simplify further.^{[8]

X

Research source

Example: Simplify ${displaystyle {sqrt {frac {100}{4}}}}$ .

${displaystyle {frac {100}{4}}=25}$ , so we can rewrite this as ${displaystyle {sqrt {25}}}$ .

${displaystyle 25=5^{2}}$ , so ${displaystyle {sqrt {25}}=5}$ .}
2
Rewrite the fraction as two radical expressions instead. Some people prefer this other method of solving problems like this. Rewrite the fraction so there is one root in the numerator and another in the denominator. Simplify each root separately, then simplify the fraction.^{[9]

X

Research source

Example: Simplify ${displaystyle {sqrt {frac {75}{12}}}}$ .
Rewrite this as ${displaystyle {frac {sqrt {75}}{sqrt {12}}}}$ .
Simplify the numerator: ${displaystyle {sqrt {75}}={sqrt {25times 3}}=5{sqrt {3}}}$

Simplify the denominator: ${displaystyle {sqrt {12}}={sqrt {4times 3}}=2{sqrt {3}}}$

Plug these back into the fraction: ${displaystyle {frac {5{sqrt {3}}}{2{sqrt {3}}}}}$

Cancel out ${displaystyle {sqrt {3}}}$ to get ${displaystyle {frac {5}{2}}}$ .}
3
Adjust your answer so there are no roots in the denominator. Sometimes, the simplest form still has a radical expression. That’s fine, but most math teachers want you to keep any radicals in the top of the fraction, not the denominator. Follow the rules for multiplying fractions to cancel out any roots on the bottom of your fraction:^{[10]

X

Research source

Example: You’ve simplified a fraction and got the answer ${displaystyle {frac {4}{9{sqrt {5}}}}}$ .
To put it in standard form, multiply the top and bottom of the fraction by the root: ${displaystyle {frac {4times {sqrt {5}}}{9{sqrt {5}}times {sqrt {5}}}}={frac {4{sqrt {5}}}{9times 5}}={frac {4{sqrt {5}}}{45}}}$}

Method 4

Method 4 of 5:

Combining Roots of Different Kinds

1
Convert roots to fractional exponents. You can rewrite any root as an exponent with a fractional value. The pattern is pretty straightforward once you’re used to it:^{[11]

X

Research source

${displaystyle {sqrt {x}}=x^{frac {1}{2}}}$
${displaystyle {sqrt[{3}]{x}}=x^{frac {1}{3}}}$
${displaystyle {sqrt[{4}]{x}}=x^{frac {1}{4}}}$
…and so on.}
2
Combine the terms using exponent rules. Once you’ve converted your terms to exponent form, follow the rules of exponents to combine them into a single expression.^{[12]

X

Research source

Example: Write ${displaystyle {sqrt {3}}{sqrt[{4}]{3}}}$ as one radical expression.
Rewrite each term in exponent form: ${displaystyle {sqrt {3}}}$ becomes ${displaystyle 3^{frac {1}{2}}}$ , and ${displaystyle {sqrt[{4}]{3}}}$ becomes ${displaystyle 3^{frac {1}{4}}}$ .
The whole expression is now ${displaystyle 3^{frac {1}{2}}times 3^{frac {1}{4}}}$ .
Since the exponents have the same base (3), multiplying them together gives us the same base raised to the sum of the two exponents: ${displaystyle 3^{({frac {1}{2}}+{frac {1}{4}})}}$ .
Simplify to ${displaystyle 3^{frac {3}{4}}}$ .}
3
Convert back to radical form. Once you have a single term with a fractional exponent, rewrite it as a radical expression. (The denominator moves to the root, and the numerator stays as an exponent.)^{[13]

X

Research source

${displaystyle 3^{frac {3}{4}}={sqrt[{4}]{3^{3}}}}$}
4
Simplify if possible. If you have any multiplication or exponents left, calculate them so your final answer is in simplest form.
- ${displaystyle {sqrt[{4}]{3^{3}}}={sqrt[{4}]{9}}}$

Method 5

Method 5 of 5:

Simplifying Radical Expressions with Variables

1
Cancel out exponents and roots just as you would with integers. Algebraic problems involve variables like ${displaystyle x}$ $x$ that can represent any number. The rules for exponents and roots still apply to these variables.^{[14]

X

Research source

Example: Simplify ${displaystyle {sqrt[{3}]{x^{2}times x}}}$ .
Combine the terms under the cube root just like you would a number: ${displaystyle {sqrt[{3}]{x^{3}}}}$

Since the root and the exponent values match, they cancel out to make ${displaystyle x}$ .}
2
Give positive solutions to even roots. Notice how ${displaystyle (-2)^{2}}$ ${displaystyle (-2)^{2}}$ and ${displaystyle 2^{2}}$ $2^{{2}}$ both equal 4. That means that 4 (or any positive number) actually has two square roots: one positive number and one negative. But the square root symbol ${displaystyle {sqrt {}}}$ ${sqrt {}}$ , by default, only refers to the positive number. For instance, you can simplify ${displaystyle {sqrt {4}}}$ ${sqrt {4}}$ to ${displaystyle 2}$ ${displaystyle 2}$ , and ignore the negative solution.^{[15]

X

Research source

The same is true of any even root: ${displaystyle {sqrt {}},{sqrt[{4}]{}},{sqrt[{6}]{}}}$ , and so on.
This does not apply to odd roots like ${displaystyle {sqrt[{3}]{}}}$ or ${displaystyle {sqrt[{5}]{}}}$ . The odd root of a negative number is always negative, and the odd root of a positive number is always positive. (Test this yourself by calculating ${displaystyle 2^{3}}$ and ${displaystyle (-2)^{3}}$ .)}
3
Use the absolute value symbol to make a variable positive. Variables are tricky: we don’t know whether they represent a positive or a negative number. Since the square root (or any even root) function must always give a positive answer, we make sure this happens by using the absolute value symbol around the answers, like this: |x|. This symbol just means “make this value positive.”^{[16]

X

Research source

Example: Simplify ${displaystyle {sqrt {32x^{2}}}}$ .
Simplify the non-variable term: ${displaystyle {sqrt {32}}={sqrt {2^{2}times 2^{2}times 2}}=(2times 2){sqrt {2}}=4{sqrt {2}}}$ .
Simplify the variable component by canceling out the root and exponent: ${displaystyle {sqrt {x^{2}}}=x}$ .
To make sure the solution to the root is positive, add absolute value symbols around that term: |x|.
Write the whole expression: 4|x| ${displaystyle {sqrt {2}}}$ .}

Video
.

By using this service, some information may be shared with YouTube.

Tips

You can find online tools or apps that will simplify a radical expression for you. You can use these to check your work.

⧼thumbs_response⧽

Helpful
0

Not Helpful
0
For complicated problems, you might need to use more than one of these methods. (For instance, you might first multiply a square root with a cube root, then simplify further, then simplify a fraction.) Make “easy” simplifications as you go (for instance, 4/2=2 or 3×5=15) and you’ll have an easier time.

⧼thumbs_response⧽

Helpful
0

Not Helpful
0

Submit a Tip

All tip submissions are carefully reviewed before being published

Submit

Thanks for submitting a tip for review!

Warnings

The square root (or any even root) of a negative number can’t be simplified without using complex numbers. For now, leave expressions like ${displaystyle {sqrt {-4}}}$ alone, and treat them as separate from the rest of the expression. (Odd roots like ${displaystyle {sqrt[{3}]{}}}$ or ${displaystyle {sqrt[{5}]{}}}$ can handle negative numbers just fine.)^{[17]

X

Research source}

⧼thumbs_response⧽

Helpful
0

Not Helpful
0