3.01 Square roots of perfect squares

Perfect squares

As we have seen previously, perfect squares are important numbers as their square root is an integer. We have actually only looked a positive square roots before. However, we can also find the negative square root of a number.

Exploration

Use the applet below to refresh your memory of the first $20$20 perfect squares. Consider the questions below as you explore.

1. Which of these perfect squares are you already familiar with?
2. How can you create a definition for a perfect square, using tiles?
3. Where might you see these numbers come up outside of math class?
4. What is the relationship between the area (number of tiles) and side length?
5. Why do you think they call multiplying a number by itself as squaring a number?

Created with Geogebra

Finding the square root

If we are asked to find the square root of a value, we are being asked, "What number multiplied by itself would give this value?"

You might also see the square root symbol written with a number inside it, for example, $\sqrt{25}$√25.

This means find the square root of $25$25.

Worked example

Question 1

Evaluate $\sqrt{144}$√144.

Think: We should read $\sqrt{144}$√144 as "the square root of $144$144".

This is the number that squares (multiplies with itself) to make $144$144.

We know that $144=12\times12$144=12×12.

Do: The square root of $144$144 is $12$12, so $\sqrt{144}=12$√144=12.

 

Question 2

Evaluate: what numbers, when squared, give $25$25?

Think: We might think of this as the same thing as $\sqrt{25}$√25, but notice that the question says numbers, not number, so we are looking for more than one. We should remember that the product of two positives is positive, but the product of two negatives is also positive. 

We know that $25=5\times5$25=5×5 and also that $25=-5\times\left(-5\right)$25=−5×(−5).

Do: The two number which when squared equal $25$25 are $5$5 and $-5$−5. 

Reflect: We can always check by doing $5^2$52 and $\left(-5\right)^2$(−5)2 to confirm that we do get back $25$25.

 

Practice questions

QUESTION 3

QUESTION 4