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iGCSE (2021 Edition)

5.10 Binomial expansions with surds (Extended)

Worksheet
Binomial products
1

Expand the following brackets:

a

\left(5 - \sqrt{13}\right) \left(5 + \sqrt{13}\right)

b

\left(\sqrt{11} - 11\right) \left(\sqrt{11} + 11\right)

c

\left( 8 \sqrt{5} - 6\right) \left( 8 \sqrt{5} + 6\right)

d

\left(\sqrt{7} + \sqrt{5}\right) \left(\sqrt{7} - \sqrt{5}\right)

e

\left( 7 \sqrt{11} - \sqrt{7}\right) \left( 7 \sqrt{11} + \sqrt{7}\right)

f

\left( 5 \sqrt{3} + 3 \sqrt{5}\right) \left( 5 \sqrt{3} - 3 \sqrt{5}\right)

2

Expand the following brackets:

a

\left(\sqrt{3} - 13\right)^{2}

b

\left(\sqrt{7} + \sqrt{3}\right)^{2}

c

\left( 3 \sqrt{3} + 8\right)^{2}

d

\left( 4 \sqrt{2} - \sqrt{13}\right)^{2}

e

\left( 3 \sqrt{2} + 4 \sqrt{13}\right)^{2}

f

\left(5 \sqrt{2} - \sqrt{32} \right)^{2}

3

Expand the following brackets:

a

\left(\sqrt{11} + 10\right) \left(\sqrt{3} - 9\right)

b

\left(\sqrt{11} - \sqrt{13}\right) \left(\sqrt{7} - \sqrt{2}\right)

c

\left( 11 \sqrt{2} - \sqrt{7}\right) \left( 13 \sqrt{3} - \sqrt{5}\right)

d

\left( 4 \sqrt{2} - \sqrt{7}\right) \left( 3 \sqrt{3} + \sqrt{8}\right)

e

\left( 17 \sqrt{3} - 8\sqrt{8}\right) \left(\sqrt{24} - \sqrt{5}\right)

f

\left( \sqrt{90} - 7\sqrt{7}\right) \left(\sqrt{72} - 6\sqrt{5}\right)

Equivalent expressions
4

Consider the following equation:

\left( 3 \sqrt{35} - 2 \sqrt{7}\right)^{2} = x - y \sqrt{5}

a

Expand and simplify the left hand side of the equation.

b

State the values of x and y.

5

Consider the following equation:

\left(\sqrt{m} + n\right)^{2} = 16 + 6 \sqrt{7}

a

Expand and simplify the left hand side of the equation.

b

State the values of m and n.

6

Consider the following expression:

\left(\sqrt{a} + 7\right)^{2} = 54+b \sqrt{5}

a

Expand and simplify the left hand side of the equation.

b

State the values of a and b.

7

Consider the following expression:

\left(x\sqrt{y} + 5\right)^{2} = 33+ 4\sqrt{2}

a

Expand and simplify the left habd side of the equation.

b

State the values of x and y.

8

Consider the following expression:

\left(\sqrt{7} + 4\right)^{2} + \left(\sqrt{7} + m\right)^{2}

a

Expand and simplify the expression.

b

Hence what value of m can be substituted into \left(\sqrt{7} + 4\right)^{2} + \left(\sqrt{7} + m\right)^{2} so that it has a rational value?

Applications
9

Consider the following right triangle:

a

Find c, the length of the hypotenuse of the triangle.

b

Find the exact perimeter of the figure.

c

Find the exact area of the figure.

10

Consider the following isosceles triangle:

a

Find h, the height of the isosceles triangle.

b

Find the exact perimeter of the triangle.

c

Find the exact area of the triangle.

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Outcomes

0607E1.10A

Surds (radicals), simplification of square root expressions.

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