Question Video: Finding the Square Root of a Negative Number Mathematics

Express √−54 in terms of 𝑖.

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Video Transcript

Express the square root of negative 54 in terms of 𝑖.

We recall that any radical or surd can be split up using our knowledge of factor pairs and products. The square root of 𝑎𝑏 is equal to the square root of 𝑎 multiplied by the square root of 𝑏. This means that the square root of negative 54 can be rewritten as the square root of 54 multiplied by the square root of negative one.

From our knowledge of complex and imaginary numbers, we know that the square root of negative one is equal to 𝑖. Nine multiplied by six is equal to 54. Therefore, the square root of 54 can be rewritten as the square root of nine multiplied by the square root of six. As the square root of nine is three, this in turn could be rewritten as three multiplied by the square root of six. As the square root of 54 is equal to three root six, the square root of negative 54 is equal to three root six 𝑖.

This leads us to a general rule that if the square root of 𝑚 is equal to 𝑛, then the square root of negative 𝑚 is equal to 𝑛 multiplied by 𝑖.