How to Multiply Square Roots

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This article was co-written by David Jia. David Jia is a tutoring teacher and founder of LA Math Tutoring, a private tutoring facility based in Los Angeles, California. With over 10 years of teaching experience, David teaches a wide variety of subjects to students of all ages and grades, as well as college admissions counseling and prep for SAT, ACT, ISEE, etc. scoring 800 in math and 690 in English on the SAT, David was awarded a Dickinson Scholarship to the University of Miami, where he graduated with a bachelor’s degree in business administration. Additionally, David has worked as an instructor in online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

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Multiplying the square root, a common root form, is similar to multiplying a regular integer. Sometimes the square root comes with a coefficient (an integer preceded by the root sign), but this factor only costs you one more multiplication. The hardest part about multiplying square roots is in the step of minimizing the result, but if you know the perfect squares things will be very simple.

Table of Contents

Multiply square root without coefficients

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Multiply the base numbers together. The base number is the number below the radical sign. ^{[1] X Research Source} When multiplying the root numbers together, we do the same as for integers. Remember to include the square in the result. ^{[2] X Research Source}

For example, when calculating $15$ $\times$ $5$ ${displaystyle {sqrt {15}}times {sqrt {5}}}$ ${sqrt {15}}times {sqrt {5}}$ , we take $15$ $\times$ $5$ $=$ $75$ ${displaystyle 15times 5=75}$ $15times 5=75$ . So, $15$ $\times$ $5$ $=$ $75$ ${displaystyle {sqrt {15}}times {sqrt {5}}={sqrt {75}}}$ ${sqrt {15}}times {sqrt {5}}={sqrt {75}}$ .

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Write the part under the radical sign as the product of a square with an integer value. To perform this step, we need to determine if the root number is a multiple of a perfect square. ^{[3] X Research Source} If it is not possible to derive a perfect square from the square root, then the result is already in a simplified form and we do not need to do any further calculations.

A perfect square is the result of multiplying a positive or negative integer by itself. ^{[4] X Research Source} For example, 25 is a perfect square because $5$ $\times$ $5$ $=$ $25$ ${displaystyle 5times 5=25}$ $5times 5=25$ .
Example with $75$ ${displaystyle {sqrt {75}}}$ ${sqrt {75}}$ , we can separate out a perfect square of 25:
$75$ ${displaystyle {sqrt {75}}}$ ${sqrt {75}}$
= $25$ $\times$ $3$ ${displaystyle {sqrt {25times 3}}}$ ${sqrt {25times 3}}$

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Set the square root of the grouped square outside the root sign. Leave the remainder under the radical mark. Here we have simplified the radical expression.

For example, $75$ ${displaystyle {sqrt {75}}}$ ${sqrt {75}}$ can be decomposed into $25$ $\times$ $3$ ${displaystyle {sqrt {25times 3}}}$ ${sqrt {25times 3}}$ , grouping the square root of 25 (which is 5) outside the root sign, we get:
$75$ ${displaystyle {sqrt {75}}}$ ${sqrt {75}}$
= $25$ $\times$ $3$ ${displaystyle {sqrt {25times 3}}}$ ${sqrt {25times 3}}$
= $5$ $3$ ${displaystyle 5{sqrt {3}}}$ $5{sqrt {3}}$

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Square root square. Sometimes we need to square a square root, or in other words, multiply that square root by itself. Square and root a number are two opposite operations; so to unsign the square root, we can square itself. The result of this operation is the number under the radical sign. ^{[5] X Research Sources}

Example: Because $25$ $\times$ $25$ $=$ $5$ $\times$ $5$ $=$ $25$ ${displaystyle {sqrt {25}}times {sqrt {25}}=5times 5=25}$ ${sqrt {25}}times {sqrt {25}}=5times 5=25$ should $25$ $\times$ $25$ $=$ $25$ ${displaystyle {sqrt {25}}times {sqrt {25}}=25}$ ${sqrt {25}}times {sqrt {25}}=25$ .

Multiply square root with coefficient

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Multiply the coefficients together. The coefficient of the square root is the number outside the root sign. To multiply the coefficients together, we just need to do the normal multiplication without considering the part with the radical sign. The product of this multiplication is preceded by the first root sign.

Pay attention to the sign (negative, positive) when multiplying the coefficient. Don’t forget the rule that the product of a negative number and a positive number is a negative number, and the product of two negative numbers is a positive number.
For example, when calculating $3$ $2$ $\times$ $2$ $6$ ${displaystyle 3{sqrt {2}}times 2{sqrt {6}}}$ $3{sqrt {2}}times 2{sqrt {6}}$ , first we need to calculate $3$ $\times$ $2$ $=$ $6$ ${displaystyle 3times 2=6}$ $3times 2=6$ . The problem becomes $6$ $2$ $\times$ $6$ ${displaystyle 6{sqrt {2}}times {sqrt {6}}}$ $6{sqrt {2}}times {sqrt {6}}$ .

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Multiply the part under the radical sign. As mentioned in the previous section, we only need to multiply the part below the radical sign as with ordinary integers. Always remember to write the product of the number obtained under the radical sign.

For example, consider $6$ $2$ $\times$ $6$ ${displaystyle 6{sqrt {2}}times {sqrt {6}}}$ $6{sqrt {2}}times {sqrt {6}}$ , to calculate the product of the lower part of the root, we take $2$ $\times$ $6$ $=$ $twelfth$ ${displaystyle 2times 6=12}$ $2times 6=12$ , Okay $2$ $\times$ $6$ $=$ $twelfth$ ${displaystyle {sqrt {2}}times {sqrt {6}}={sqrt {12}}}$ ${sqrt {2}}times {sqrt {6}}={sqrt {12}}$ . The problem becomes $6$ $twelfth$ ${displaystyle 6{sqrt {12}}}$ $6{sqrt {12}}$ .

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Simplify the lower part of the product root of the perfect square. This step will help us shorten the answer. ^{[6] X Research Source} If we can’t extract from the lower part the root of a perfect square, then the calculated answer is simplified and we can stop the calculation here.

A perfect square is the result of multiplying an integer (negative or positive) by itself. ^{[7] X Research Source} For example, 4 is a perfect square because $2$ $\times$ $2$ $=$ $4$ ${displaystyle 2times 2=4}$ $2times 2=4$ .
For example, the word $twelfth$ ${displaystyle {sqrt {12}}}$ ${sqrt {12}}$ we can separate 4 from the lower part
$twelfth$ ${displaystyle {sqrt {12}}}$ ${sqrt {12}}$
= $4$ $\times$ $3$ ${displaystyle {sqrt {4times 3}}}$ ${sqrt {4times 3}}$

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Multiply the square root of the squared square by the coefficient. The rest put under the radical sign, we will get the reduced result of the calculation.

For example, $6$ $twelfth$ ${displaystyle 6{sqrt {12}}}$ $6{sqrt {12}}$ can be decomposed into $6$ $4$ $\times$ $3$ ${displaystyle 6{sqrt {4times 3}}}$ $6{sqrt {4times 3}}$ , take the square root of 4 (which is 2) out of the root sign and multiply this number by a factor of 6:
$6$ $twelfth$ ${displaystyle 6{sqrt {12}}}$ $6{sqrt {12}}$
= $6$ $4$ $\times$ $3$ ${displaystyle 6{sqrt {4times 3}}}$ $6{sqrt {4times 3}}$
= $6$ $\times$ $2$ $3$ ${displaystyle 6times 2{sqrt {3}}}$ $6times 2{sqrt {3}}$
= $twelfth$ $3$ ${displaystyle 12{sqrt {3}}}$ $12{sqrt {3}}$

Advice

Try to remember the value of the squares, so the calculation with square roots will be much easier.
Follow the sign rules to determine whether the new coefficient has a positive or negative sign. One positive coefficient multiplied by another negative will get a negative coefficient. The product of two coefficients with the same sign will result in a positive coefficient.
All the parts under the radical sign must be positive, so when you multiply the parts under the radical sign together you don’t need to care about their sign.

Things you need

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Steps

Multiply square root without coefficients

Multiply square root with coefficient

Advice

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