How To Find The Square Root of a Negative Number

TL;DR

The square root of a negative number is simplified by using the imaginary number i, which equals the square root of negative one. For example, the square root of negative 9 becomes the square root of 9 times i, or 3i. For non-perfect squares, factor out the highest perfect square. Read on for worked examples with negative 49, negative 75, negative 108, negative 80, and a quadratic equation.

Transcript

how would you simplify this expression what is the square root of negative 9. how can we find the square root of a negative number let's compare this to the square root of 9. the square root of 9 is 3 because we know that 3 times 3 is equal to 9. but now if we were to multiply negative 3 together that will also give us positive 9. we can't find two... Read More

Key Insights

• ❎ When simplifying square root expressions with negative numbers, introduce imaginary numbers.
• ❎ The square root of a negative number can be expressed as the square root of the positive value times the square root of negative one.
• ❎ In simplifying square root expressions with negative numbers, identify the highest perfect square that goes into the given number.
• ❎ Imaginary numbers allow us to find the square root of negative numbers and solve quadratic equations.
• ❎ When taking the square root of x squared, there are two possible answers: the positive and negative square root of x.

Questions & Answers

Q: How do you find the square root of a negative number?

Introduce the imaginary number i, which equals the square root of negative one. Separate the negative radicand into its positive value times negative one, then simplify the positive square root and multiply it by i.

Q: What is the square root of negative 9?

The square root of negative 9 is 3i. It can be written as the square root of 9 times the square root of negative one, which simplifies to 3 times i.

Q: Why is the square root of negative 9 not 3 or negative 3?

Three times three equals positive 9, and negative 3 times negative 3 also equals positive 9. Neither pair of identical numbers produces negative 9, so the imaginary number i is needed.

Q: What is the square root of negative 49?

The square root of negative 49 simplifies to 7i. The square root of 49 is 7, while the square root of negative one is i.

Q: How do you simplify the square root of negative 75?

Factor 75 as 25 times 3 because 25 is the highest perfect square that divides 75. The expression then simplifies to 5 square root 3 times i.

Q: How do you simplify the square root of negative 108?

Factor 108 as 36 times 3 because 36 is the highest perfect square that divides 108. The square root of negative 108 therefore simplifies to 6 square root 3 times i.

Q: How do you simplify negative square root of negative 80?

Factor 80 as 16 times 5, then keep the negative sign outside the square root. Since the square root of 16 is 4 and the square root of negative one is i, the result is negative 4 square root 5 times i.

Q: How do you solve x squared plus 18 equals zero?

Subtract 18 to get x squared equals negative 18, then take the square root of both sides and include both possible signs. Factoring 18 as 9 times 2 gives x equals plus or minus 3 square root 2 times i.

Summary & Key Takeaways

• When simplifying square root expressions with negative numbers, introduce imaginary numbers.

• The square root of negative 9 is equal to 3i.

• The highest perfect square that goes into a given number can be used to simplify square root expressions with negative numbers.