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This icon indicates the slide contains activities created in Flash. These activities are not editable. This icon indicates teacher’s notes in the Notes field. For more detailed instructions, see the Getting Started presentation. 1 of 25 © Boardworks Ltd 2011

Square numbers When we multiply a number by itself, we say that we are

Square numbers When we multiply a number by itself, we say that we are squaring the number. To square a number, we can write a small 2 after it. For example, the number 3 multiplied by itself can be written as: 3× 3 or 32 The value of three squared is 9. The result of any whole number multiplied by itself is called a square number. 2 of 25 © Boardworks Ltd 2011

Square roots Finding the square root of a number is the opposite action of

Square roots Finding the square root of a number is the opposite action of finding the square. It is the inverse operation. squared 8 64 square rooted When we find the square root of a number, we write: √ 64 = 8. This is the same as saying ‘the square root of 64 is 8’. 3 of 25 © Boardworks Ltd 2011

The product of two square numbers This student has a theory that he thinks

The product of two square numbers This student has a theory that he thinks is true. He says that: ‘the product of two square numbers is always another square number’. 4 × 25 = 100, 9 × 4 = 36 and 16 × 4 = 64. However, he is struggling to prove that his theory is correct. Do you think he is correct? Can you show that it is correct for the examples above? Can you prove it algebraically? 4 of 25 © Boardworks Ltd 2011

Square number products 5 of 25 © Boardworks Ltd 2011

Square number products 5 of 25 © Boardworks Ltd 2011

Using factors to find square roots If a number has factors that are square

Using factors to find square roots If a number has factors that are square numbers then we can use these factors to find the square root. Can you find 400 ? Can you find 225 ? √ 400 = √(4 × 100) √ 225 = √(9 × 25) = √ 4 × √ 100 = √ 9 × √ 25 = 2 × 10 =3× 5 = 20 = 15 How would you go about finding 3969 ? 6 of 25 © Boardworks Ltd 2011

Finding square roots of decimals We can also find the square root of a

Finding square roots of decimals We can also find the square root of a small number when we divide two square numbers. Can you find 0. 09 ? Can you find 0. 0144 ? 0. 09 = (9 ÷ 100) 0. 0144 = (144 ÷ 10000) = √ 9 ÷ √ 100 = √ 144 ÷ √ 10000 = 3 ÷ 10 = 12 ÷ 100 = 0. 3 = 0. 12 How would you go about finding 0. 16 ? 7 of 25 © Boardworks Ltd 2011

Finding square roots 8 of 25 © Boardworks Ltd 2011

Finding square roots 8 of 25 © Boardworks Ltd 2011

Approximate square roots If a number cannot be written as a product or quotient

Approximate square roots If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly. What is the 2 ? If you enter this on your calculator, it shows as 1. 414213562 (to 9 decimal places). The number of digits after the decimal point is infinite and non-repeating. This is an example of an irrational number. What other irrational numbers can you think of? 9 of 25 © Boardworks Ltd 2011

Estimating square roots When trying to find the square root of a number that

Estimating square roots When trying to find the square root of a number that is not a square number, we can estimate the value using our knowledge of square roots. What is 50 ? We know that 50 lies between 49 and 64. Therefore: 49 < 50 < 64. As a result, we can say that: 7 < 50 < 8. 50 is much closer to 49 than to 64, so 50 will be about 7. 1. Using a calculator, we can find a more exact value of √ 50. 50 = 7. 07 (to 2 decimal places. ) 10 of 25 © Boardworks Ltd 2011

An accurate estimate? 11 of 25 © Boardworks Ltd 2011

An accurate estimate? 11 of 25 © Boardworks Ltd 2011

Negative square roots When finding the square root of a number, remember the special

Negative square roots When finding the square root of a number, remember the special relationship between positive and negative numbers. Can you solve the equation x 2 = 25? The equation x² = 25 has two solutions: x = 5 and x = – 5. Therefore, the square root of 25 is 5 or – 5. When we use the symbol we usually mean the positive square root. We also write ± to mean both the positive and the negative square root. 12 of 25 © Boardworks Ltd 2011

Squares and square roots from a graph 13 of 25 © Boardworks Ltd 2011

Squares and square roots from a graph 13 of 25 © Boardworks Ltd 2011

Cubes When we multiply a number by itself and then by itself again, we

Cubes When we multiply a number by itself and then by itself again, we say that we are cubing the number. 13 = 1 × 1 = 1 ‘ 1 cubed’ or ‘ 1 to the power of 3’ 23 = 2 × 2 = 8 ‘ 2 cubed’ or ‘ 2 to the power of 3’ 33 = 3 × 3 = 27 ‘ 3 cubed’ or ‘ 3 to the power of 3’ What is the value of 4³? Show your working. What is the value of 5³? Show your working. 14 of 25 © Boardworks Ltd 2011

Cube recognition 15 of 25 © Boardworks Ltd 2011

Cube recognition 15 of 25 © Boardworks Ltd 2011

Cube roots Finding the cube root is the opposite action to finding the cube

Cube roots Finding the cube root is the opposite action to finding the cube value of a number. It is the inverse operation. cubed 5 125 cube rooted When we find the cube root of a number, we write: 3√ 125 = 5. This is the same as saying ‘the cube root of 125 is 5’. 16 of 25 © Boardworks Ltd 2011

Squares, cubes and roots 17 of 25 © Boardworks Ltd 2011

Squares, cubes and roots 17 of 25 © Boardworks Ltd 2011

Index notation We use index notation to show repeated multiplication by the same number.

Index notation We use index notation to show repeated multiplication by the same number. We can use index notation to simplify the way we write the expression: 2 × 2 × 2. Base 25 Index or power This number is read as ‘two to the power of five’. 25 = 2 × 2 × 2 = 32 Use index notation to show: 2 × 2 × 2. What is the value of the calculation? 18 of 25 © Boardworks Ltd 2011

Index notation 19 of 25 © Boardworks Ltd 2011

Index notation 19 of 25 © Boardworks Ltd 2011

A justifiable conclusion? Two students in Miss Jones’ class notice an interesting pattern emerging

A justifiable conclusion? Two students in Miss Jones’ class notice an interesting pattern emerging when they are raising negative numbers to different powers. When we raise a negative number to an odd power, the answer is negative. When we raise a negative number to an even power the answer is positive. Using your knowledge of how positive and negative numbers interact when multiplied, can you prove that this is always true? 20 of 25 © Boardworks Ltd 2011

Index laws 21 of 25 © Boardworks Ltd 2011

Index laws 21 of 25 © Boardworks Ltd 2011

Using index laws 22 of 25 © Boardworks Ltd 2011

Using index laws 22 of 25 © Boardworks Ltd 2011

The power of 1 Raising any number to the power of 1 produces some

The power of 1 Raising any number to the power of 1 produces some interesting results. 61 471 0. 91 – 51 01 What do you think the value of these calculations is? Justify your answer. Any number raised to the power of 1 is equal to the number itself. As a result, we don’t usually write the power when a number is raised to the power of 1. x 1 = x 23 of 25 © Boardworks Ltd 2011

The power of 0 As well as raising numbers to the power of 1,

The power of 0 As well as raising numbers to the power of 1, it is also possible to raise numbers to the power of 0. Consider the following calculation: 64 ÷ 64 = 1. Now look at these three calculations: 60 = 1 3. 4520 = 1 723 538 5920 = 1 How can we use the information above to show that x 0 is equal to 1? Is this always true? What happens when x = 0? 24 of 25 © Boardworks Ltd 2011

Index laws 25 of 25 © Boardworks Ltd 2011

Index laws 25 of 25 © Boardworks Ltd 2011