How to Do Square Root Times a Number

Radicals come before multiplication in the order of operations.

You can use the order of operations, which determines the order in which to solve an expression or equation, to guide you through the process of solving an expression with multiple signs and numbers. Exponents and radicals come before multiplication in the order of operations, which means you must solve the square root of a number before multiplying that number by another number. A square root is designated by a radical sign without a root or index number. The square root of a number is a number that when squared, or multiplied by itself, equals the original number. For instance, the square root of 9 is 3 because 3 times 3 equals 9.

Instructions

1. Find an expression that contains the square root of a number times a number. For example, use 5[sqrt(49)], with "sqrt" representing "square root." This expression represents 5 times the square root of 49, with the number 49 inside a radical sign.

2. Determine the square root of the number inside the radical sign as the first operation in solving the expression because radicals come before multiplication in the order of operations. In the example, find the number that, when multiplied by itself, equals 49. Multiply a whole number, such as 6, by itself to see if the result equals 49. The number 6 times 6 equals 36. This is close, but not equal, to 49, so 6 is not the square root of 49.

3. Multiply the next higher whole number by itself to see if the result equals the number inside the radical. The next whole number is 7, so multiply 7 times 7. The result is 49, which means that 7 is the square root of 49. This eliminates the radical sign and leaves 5 times 7, or the expression 5(7).

4. Multiply the remaining numbers in the expression. In the example, multiply 5 times 7. The result is 35.

Tips & Warnings

• If you determine in Steps 2 and 3 that the number inside the radical sign does not have a whole number as its square root, such as 6 or 7, use the square root button on a calculator to calculate the square root of the number inside the radical.

• If the number inside the square root sign is negative, the expression has no real number solution because a number multiplied by itself is always positive.

References

• West Texas A&M University; Operations on Real Numbers; Kim Seward; June 2003
• Maths Is Fun: Squares and Square Roots
• Union County College: Square Roots