Example: Simplify 45{\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: 45÷3=15{\displaystyle 45\div 3=15}, so 45=3×15{\displaystyle 45=3\times 15}. 45=3×15{\displaystyle {\sqrt {45}}={\sqrt {3\times 15}}}
Now we have 3×15{\displaystyle {\sqrt {3\times 15}}}, but we can factor 15 again into 3×5{\displaystyle 3\times 5}. 45=3×3×5{\displaystyle {\sqrt {45}}={\sqrt {3\times 3\times 5}}}
In 3×3×5{\displaystyle {\sqrt {3\times 3\times 5}}}, the number 3 shows up twice. Since 3×3=32{\displaystyle 3\times 3=3^{2}}, we can rewrite the whole expression as 32×5{\displaystyle {\sqrt {3^{2}\times 5}}}.
32×5=325{\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. ) 325=35{\displaystyle {\sqrt {3^{2}}}{\sqrt {5}}=3{\sqrt {5}}} Since there are no other exponents left under the square root, you’re all done!
Example: Simplify 360{\displaystyle {\sqrt {360}}}. This takes a lot of factoring to break down: 360=40×9=(5×8)×(3×3)=5×2×2×2×3×3{\displaystyle {\sqrt {360}}={\sqrt {40\times 9}}={\sqrt {(5\times 8)\times (3\times 3)}}={\sqrt {5\times 2\times 2\times 2\times 3\times 3}}} Rewrite pairs of numbers using exponents: 5×22×2×32{\displaystyle {\sqrt {5\times 2^{2}\times 2\times 3^{2}}}}. Bring the 2 and 3 outside the square root: 2×3×5×2{\displaystyle 2\times 3\times {\sqrt {5\times 2}}} Simplify the numbers in front of the square root: 65×2{\displaystyle 6{\sqrt {5\times 2}}} To get the final answer, simplify the numbers under the square root: 610{\displaystyle 6{\sqrt {10}}}
Example: Simplify 813{\displaystyle {\sqrt[{3}]{81}}} (the cube root of 81). 81=3×3×3×3{\displaystyle 81=3\times 3\times 3\times 3}, so 813=3×3×3×33{\displaystyle {\sqrt[{3}]{81}}={\sqrt[{3}]{3\times 3\times 3\times 3}}}
3×3×3×33=343{\displaystyle {\sqrt[{3}]{3\times 3\times 3\times 3}}={\sqrt[{3}]{3^{4}}}}
343=33×33=33333{\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 333{\displaystyle {\sqrt[{3}]{3^{3}}}}, they cancel out, leaving only the base number, 3{\displaystyle 3}. Plug that into the whole expression to get 333{\displaystyle 3{\sqrt[{3}]{3}}}. Since there are no more exponents left that can cancel out, this is the simplified form.
Example: Simplify 27×35×75{\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 27535575{\displaystyle {\sqrt[{5}]{2^{7}}}{\sqrt[{5}]{3^{5}}}{\sqrt[{5}]{7}}}. 275=25×225{\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 2225{\displaystyle 2{\sqrt[{5}]{2^{2}}}} In 355{\displaystyle {\sqrt[{5}]{3^{5}}}}, the root and exponent cancel out to make 3{\displaystyle 3}. Since 75{\displaystyle {\sqrt[{5}]{7}}} has no exponents that cancel out, this can’t be simplified. Plug your simplified terms back into the whole expression: 2225×3×75{\displaystyle 2{\sqrt[{5}]{2^{2}}}\times 3\times {\sqrt[{5}]{7}}} Combine like terms: (2×3)22×75{\displaystyle (2\times 3){\sqrt[{5}]{2^{2}\times 7}}} Calculate multiplication and exponents: 6285{\displaystyle 6{\sqrt[{5}]{28}}}
Example: Simplify 1004{\displaystyle {\sqrt {\frac {100}{4}}}}. 1004=25{\displaystyle {\frac {100}{4}}=25}, so we can rewrite this as 25{\displaystyle {\sqrt {25}}}. 25=52{\displaystyle 25=5^{2}}, so 25=5{\displaystyle {\sqrt {25}}=5}.
Example: Simplify 7512{\displaystyle {\sqrt {\frac {75}{12}}}}. Rewrite this as 7512{\displaystyle {\frac {\sqrt {75}}{\sqrt {12}}}}. Simplify the numerator: 75=25×3=53{\displaystyle {\sqrt {75}}={\sqrt {25\times 3}}=5{\sqrt {3}}} Simplify the denominator: 12=4×3=23{\displaystyle {\sqrt {12}}={\sqrt {4\times 3}}=2{\sqrt {3}}} Plug these back into the fraction: 5323{\displaystyle {\frac {5{\sqrt {3}}}{2{\sqrt {3}}}}} Cancel out 3{\displaystyle {\sqrt {3}}} to get 52{\displaystyle {\frac {5}{2}}}.
Example: You’ve simplified a fraction and got the answer 495{\displaystyle {\frac {4}{9{\sqrt {5}}}}}. To put it in standard form, multiply the top and bottom of the fraction by the root: 4×595×5=459×5=4545{\displaystyle {\frac {4\times {\sqrt {5}}}{9{\sqrt {5}}\times {\sqrt {5}}}}={\frac {4{\sqrt {5}}}{9\times 5}}={\frac {4{\sqrt {5}}}{45}}}
x=x12{\displaystyle {\sqrt {x}}=x^{\frac {1}{2}}} x3=x13{\displaystyle {\sqrt[{3}]{x}}=x^{\frac {1}{3}}} x4=x14{\displaystyle {\sqrt[{4}]{x}}=x^{\frac {1}{4}}} . . . and so on.
Example: Write 334{\displaystyle {\sqrt {3}}{\sqrt[{4}]{3}}} as one radical expression. Rewrite each term in exponent form: 3{\displaystyle {\sqrt {3}}} becomes 312{\displaystyle 3^{\frac {1}{2}}}, and 34{\displaystyle {\sqrt[{4}]{3}}} becomes 314{\displaystyle 3^{\frac {1}{4}}}. The whole expression is now 312×314{\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: 3(12+14){\displaystyle 3^{({\frac {1}{2}}+{\frac {1}{4}})}}. Simplify to 334{\displaystyle 3^{\frac {3}{4}}}.
334=334{\displaystyle 3^{\frac {3}{4}}={\sqrt[{4}]{3^{3}}}}
334=94{\displaystyle {\sqrt[{4}]{3^{3}}}={\sqrt[{4}]{9}}}
Example: Simplify x2×x3{\displaystyle {\sqrt[{3}]{x^{2}\times x}}}. Combine the terms under the cube root just like you would a number: x33{\displaystyle {\sqrt[{3}]{x^{3}}}} Since the root and the exponent values match, they cancel out to make x{\displaystyle x}.
The same is true of any even root: ,4,6{\displaystyle {\sqrt {}},{\sqrt[{4}]{}},{\sqrt[{6}]{}}}, and so on. This does not apply to odd roots like 3{\displaystyle {\sqrt[{3}]{}}} or 5{\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 23{\displaystyle 2^{3}} and (−2)3{\displaystyle (-2)^{3}}. )
Example: Simplify 32x2{\displaystyle {\sqrt {32x^{2}}}}. Simplify the non-variable term: 32=22×22×2=(2×2)2=42{\displaystyle {\sqrt {32}}={\sqrt {2^{2}\times 2^{2}\times 2}}=(2\times 2){\sqrt {2}}=4{\sqrt {2}}}. Simplify the variable component by canceling out the root and exponent: x2=x{\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|2{\displaystyle {\sqrt {2}}}.
