Square Roots

Square Roots – Definition

Square roots are a fundamental concept in mathematics, especially in algebra and geometry. They are used in various real-life scenarios, such as calculating the area of land, determining the size of a television screen or in architectural designs.

The Square Root Symbol

There are two important terms to remember when you’re dealing with square roots. They are the radical sign and the radicand.

Understanding Square Roots

The concept of square roots is crucial for working with the Pythagorean theorem, understanding geometric shapes and dealing with powers and exponents.

The following concepts are helpful when dealing with square roots:

• Perfect squares: Numbers like $9$, $16$ or $25$ are perfect squares because they can be expressed as the square of an integer.
• Irrational numbers: Square roots of non-perfect squares, like $\sqrt{2}$ or $\sqrt{17}$, are irrational numbers. They cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal representations. Here is some more information on Rational and Irrational Numbers.
• Positive and negative roots: Every positive number has two square roots - a positive and a negative root. For instance, both $+3$ and $-3$ are square roots of $9$.

Perfect Squares – Chart

The table below shows an overview of common perfect squares:

It is helpful to memorise the perfect squares up until at least $12^2=144$, especially when you progress to more challenging concepts.

Square Roots – Guided Practice

Let’s work through the following question together: Find the square root of $49$.

Calculating Square Roots Using Prime Factorisation

When you're finding the square root of a number, especially a large one, prime factorisation is a useful tool. This method involves breaking down a number into its prime factors – the prime numbers that multiply together to make the original number. Once you have the prime factors, you can easily find the square root, especially for perfect squares.

Example 1

• The prime factors of $144$ are $2 \times 2 \times2 \times 2 \times 3\times 3$.
• These can be paired into three total groups: $2 \times 2 \times 3$.
• $2 \times 2 \times 3 = 12$ so the square root of $144$ is $12$.

Example 2

• The prime factors of $324$ are $2 \times 2 \times 3 \times 3 \times 3\times 3$
• These can be paired into the groups $2 \times 3 \times 3$
• $2 \times 3 \times 3 = 18$ so the square root of $324$ is $18$

Here is another example of a large number where it would be a good idea to use prime factorisation to find the square root.

Example 3: Find the square root of $1,089$.

• The prime factors of $1,089$ are $3 \times 3 \times 11 \times 11$
• These can be paired into the groups $3 \times 11$
• $3 \times 11 = 33$ so the square root of $1,089$ is $33$

Square Roots – Exercises

Here are some more exercises on square roots. Find the answer to the question and click on the question box to reveal the correct answer.

Square Roots – Real-World Problems

Let’s take a look at some real-world examples including square roots.

Square Roots – Summary

Let’s review what we’ve learnt about square roots in this text:

Understanding Square roots can come in handy when you learn more about the Pythagorean theorem.

Square Roots – Frequently Asked Questions