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$1$ . $1$ irrational numbers $\newline$ $10$ a Use a calculator to find $i$ $\newline$ $(\sqrt{2}+1)\times(\sqrt{2}-1)$ $\newline$ b Continue the pattern of $ii$ $\newline$ $(\sqrt{3}+1)\times(\sqrt{3}-1)$ $\newline$ c Generalise the restern of the multiplications in part a. $\newline$ d Check your the results to find $\newline$ $(\sqrt{N}+1)\times(\sqrt{N}-1)$ where $\newline$ $N$ is a positive integer. $\newline$ $11$ Here is a decim generalisation with further examples. $\newline$ Arun says:

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GCF and LCM: word problems

Full solution

Q.

1

.

1

irrational numbers

\newline

10

a Use a calculator to find

i

\newline

(\sqrt{2}+1)\times(\sqrt{2}-1)

\newline

b Continue the pattern of

ii

\newline

(\sqrt{3}+1)\times(\sqrt{3}-1)

\newline

c Generalise the restern of the multiplications in part a.

\newline

d Check your the results to find

\newline

(\sqrt{N}+1)\times(\sqrt{N}-1)

where

\newline

N

is a positive integer.

\newline

11

Here is a decim generalisation with further examples.

\newline

Arun says:

Identify pattern in multiplication: : Identify the pattern in the multiplication of expressions of the form $(\sqrt{N}+1)\times(\sqrt{N}-1)$ .
Apply difference of squares: : Apply the difference of squares formula to the expression $(\sqrt{2}+1)\times(\sqrt{2}-1)$ .
Generalize pattern for any N: : Apply the difference of squares formula to the expression (\sqrt{ $3$ }+ $1$ )\times(\sqrt{ $3$ } $-1$ ).
Verify generalization: : Generalize the pattern observed in parts a and b to any positive integer $N$ .
: : Verify the generalization by applying it to the expression $(\sqrt{N}+1)\times(\sqrt{N}-1)$ .

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