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1.08 Square roots and perfect squares

Lesson

Square numbers

We can represent repeated addition using multiplication, for example:

3+3+3+3=4\times3

In a similar way, we can represent repeated multiplication using powers.

To indicate that a number has been multiplied by itself we write the number being multiplied, called the base, and then indicate the number of times it is multiplied by itself by writing this number, called the power (also known as the index or exponent), above and to the right of the base. Here is an example:

3\times3=3^2

This is read as "three to the power of two" or "three raised to the power of 2."

For now we will just be looking at numbers raised to the power of two, known as square numbers or perfect squares. In the above example 3^2 is often referred to as "three squared".

What does it mean to 'square' a number? Look at the pattern below:

Table that contains dot formations for each squared number from 1 to 25. Ask your teacher for more information.

You can see above that each formation of dots forms a square.

\displaystyle 1^2\displaystyle =\displaystyle 1\times1=1
\displaystyle 2^2\displaystyle =\displaystyle 2\times2=4
\displaystyle 3^2\displaystyle =\displaystyle 3\times3=9
\displaystyle 4^2\displaystyle =\displaystyle 4\times4=16and so on.

Exploration

Use the applet below to explore the first 12 square numbers

Loading interactive...

The number of the squares is equal to the base squared.

Examples

Example 1

Evaluate 4^{2}+9^{2}.

Worked Solution
Create a strategy

Use the fact that squaring a number means multiplying by itself.

Apply the idea
\displaystyle 4^{2}+9^{2}\displaystyle =\displaystyle 4\times 4 + 9\times 9Multiply the base by itself
\displaystyle =\displaystyle 16 + 81Evaluate the multiplication
\displaystyle =\displaystyle 97Evaluate
Idea summary

Perfect squares can be obtained by raising a smaller number to the power of two or multiplying a number by itself.

When writing powers, the number being multiplied is called the base, the number of times it is multiplied by itself is called the power. In the example below 3 is the base and 2 is the power.

3\times3=3^2

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}.

This means find the square root of 25.

Examples

Example 2

Evaluate \sqrt{5^2+12^2}.

Worked Solution
Create a strategy

Perform the operation inside the square root symbol then find the square root of the result.

Apply the idea
\displaystyle \sqrt{5^2+12^2}\displaystyle =\displaystyle \sqrt{5 \times 5 + 12 \times 12}Multiply 5 and 12 by itself
\displaystyle =\displaystyle \sqrt{25+144}Evaluate the multiplication
\displaystyle =\displaystyle \sqrt{169}Evaluate the addition
\displaystyle =\displaystyle 13Find the square root
Reflect and check

\sqrt{169}=13 since 13^2=169.

Idea summary

A perfect square is a number that has a whole number as its square root.

To find the square root of a value, we need to find the number that when multiplied by itself would give the original value.

e.g. \sqrt{25}=5 since 5^2=25.

Outcomes

VCMNA238

Investigate index notation and represent whole numbers as products of powers of prime numbers

VCMNA239

Investigate and use square roots of perfect square numbers.

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