CTJan27 Online JMSS - Surds Multiplication

1

Expand and simplify the binomial product: $(3\sqrt{2} + 1)(\sqrt{2} - 5)$.

$1 - 14\sqrt{2}$ $11 - 14\sqrt{2}$ $1 + 16\sqrt{2}$ $6 - 15\sqrt{2}$

2

Determine the simplified form of the perfect square: $(\sqrt{7} - 3\sqrt{2})^2$.

$25 - 6\sqrt{14}$ $16 - 6\sqrt{14}$ $25 - 18\sqrt{14}$ $11 - 6\sqrt{14}$

3

Multiply the conjugate pair: $(5\sqrt{3} - \sqrt{5})(5\sqrt{3} + \sqrt{5})$.

$70$ $80$ $70 - 10\sqrt{15}$ $70 + 10\sqrt{15}$

4

Apply the distributive law and simplify: $2\sqrt{6}(3\sqrt{2} - 5\sqrt{3})$.

$12\sqrt{3} - 30\sqrt{2}$ $6\sqrt{12} - 10\sqrt{18}$ $12\sqrt{3} - 15\sqrt{2}$ $12\sqrt{3} + 30\sqrt{2}$

5

Calculate and simplify: $(\sqrt{12} + \sqrt{3})(\sqrt{3})$.

$9$ $6 + \sqrt{6}$ $3 + 2\sqrt{3}$ $3\sqrt{3} + 6$

6

Use the core multiplication rule and simplify the result: $\sqrt{8} \cdot \sqrt{18}$.

$12$ $6\sqrt{4}$ $8\sqrt{2}$ $72$

7

Factor the surd expression $3\sqrt{6} + 6\sqrt{2}$ completely.

$3\sqrt{2}(\sqrt{3} + 2)$ $3\sqrt{6}(1 + \sqrt{3})$ $\sqrt{2}(3\sqrt{3} + 6)$ $6\sqrt{2}(\frac{1}{2}\sqrt{3} + 1)$

8

Expand and simplify: $(2\sqrt{5} - 3)(4\sqrt{5} + 1)$.

$37 - 10\sqrt{5}$ $43 - 10\sqrt{5}$ $37 + 10\sqrt{5}$ $37 - 14\sqrt{5}$

9

Simplify the expression: $(2 + \sqrt{11})^2$.

$15 + 4\sqrt{11}$ $15 + 2\sqrt{11}$ $7 + 4\sqrt{11}$ $15 + 4\sqrt{22}$

10

Which of the following expressions results in a rational number?

$(\sqrt{10} + 1)^2$ $(2\sqrt{3} + 3)(\sqrt{3} - 1)$ $(\sqrt{18} - \sqrt{2})(\sqrt{8} + \sqrt{2})$ $\sqrt{5}(2\sqrt{5} + 3)$

11

Simplify the product: $(3\sqrt{24})(2\sqrt{8})$.

$48\sqrt{3}$ $12\sqrt{192}$ $24\sqrt{12}$ $96\sqrt{3}$

12

What is the conjugate of the surd expression $4 - \sqrt{32}$?

$4 + \sqrt{32}$ $4\sqrt{2} - 4$ $16 + \sqrt{32}$ $-4 + \sqrt{32}$

13

Factor the expression $15 - 5\sqrt{3}$ completely, using surd factors where applicable.

$5(3 - \sqrt{3})$ $5\sqrt{3}(\sqrt{3} - 1)$ $5\sqrt{3}(1 - \sqrt{3})$ $15\sqrt{3}(1 - \frac{1}{3})$

14

Expand and simplify: $(\sqrt{5} + \sqrt{3})(\sqrt{5} - 2\sqrt{3})$.

$-1 - \sqrt{15}$ $1 - \sqrt{15}$ $1 - 3\sqrt{15}$ $11 + \sqrt{15}$

15

Simplify the advanced perfect square expression: $(\sqrt{10} + \frac{1}{\sqrt{10}})^2$.

$\frac{121}{10}$ $10 + \frac{1}{\sqrt{10}}$ $12 + \frac{1}{\sqrt{10}}$ $\frac{101}{10}$

16

Calculate and simplify: $\sqrt{3}( \sqrt{27} + \sqrt{12})$.

$15$ $5\sqrt{9}$ $3\sqrt{9} + 6\sqrt{3}$ $9 + 6\sqrt{3}$

17

Calculate the rational result of the multiplication: $(\sqrt{20} + \sqrt{5})\sqrt{5}$.

$15$ $10 + 5\sqrt{5}$ $5\sqrt{5}$ $5 + 2\sqrt{5}$

18

If $(a\sqrt{5} + b)^2 = 29 + 12\sqrt{5}$, where $a$ and $b$ are positive integers, what is the value of $a^2 + b^2$?

$13$ $11$ $17$ $58$

19

Expand and fully simplify: $\sqrt{3}(2 + \sqrt{6} + \sqrt{12})$.

$6 + 2\sqrt{3} + 3\sqrt{2}$ $6 + 5\sqrt{3}$ $6 + 2\sqrt{3} + 6\sqrt{3}$ $6 + 2\sqrt{18} + 3\sqrt{3}$

20

Determine the product of $2\sqrt{5} - 3$ and its conjugate.

$11$ $4\sqrt{5} + 6$ $29$ $11 + 12\sqrt{5}$

21

Simplify: $3\sqrt{2}(5\sqrt{6} - 2\sqrt{2})$.

$30\sqrt{3} - 12$ $15\sqrt{8} - 8$ $30\sqrt{12} - 6$ $30\sqrt{3} + 12$

22

Expand and simplify: $(2\sqrt{5} + 3\sqrt{3})(3\sqrt{5} - \sqrt{3})$.

$21 + 7\sqrt{15}$ $39 + 7\sqrt{15}$ $21 + 11\sqrt{15}$ $39 + 11\sqrt{15}$

23

Evaluate the product of the conjugate pair: $(\sqrt{13} - 4)(\sqrt{13} + 4)$.

$-3$ $29$ $-1$ $13 - 16\sqrt{13}$

24

Expand and simplify the surd binomial squared: $(\sqrt{7} + 2\sqrt{3})^2$.

$19 + 4\sqrt{21}$ $19 + 2\sqrt{21}$ $13 + 4\sqrt{21}$ $19 + 4\sqrt{7}$

25

Simplify the expression: $\sqrt{3}(2\sqrt{6} - \sqrt{3}) + (5 - \sqrt{2})(5 + \sqrt{2})$.

$20 + 6\sqrt{2}$ $26 + 6\sqrt{2}$ $20 + 3\sqrt{6}$ $20 - 6\sqrt{2}$

26

What rational number results from simplifying $(3\sqrt{2} - \sqrt{5})(3\sqrt{2} + \sqrt{5})$?

$13$ $23$ $13 + 6\sqrt{10}$ $18 - \sqrt{25}$

27

Expand and simplify: $4\sqrt{5}(2\sqrt{10} + 3\sqrt{5})$.

$60 + 40\sqrt{2}$ $60 + 8\sqrt{50}$ $20 + 40\sqrt{2}$ $60 + 40\sqrt{10}$

28

Simplify $(5\sqrt{2} - 3)^2$.

$59 - 30\sqrt{2}$ $41 - 30\sqrt{2}$ $59 + 30\sqrt{2}$ $59 - 15\sqrt{2}$

29

Expand and simplify: $(2\sqrt{3} - \sqrt{5})(4\sqrt{3} + 3\sqrt{5})$.

$9 + 2\sqrt{15}$ $39 + 10\sqrt{15}$ $9 + 10\sqrt{15}$ $39 + 2\sqrt{15}$

30

Given the expression $E = (5\sqrt{3} - 2\sqrt{2})$. If $E$ is multiplied by its conjugate, what is the resulting rational number?

$67$ $83$ $67 - 20\sqrt{6}$ $280$

31

Simplify: $(\sqrt{5} + 1)^2 - (\sqrt{5} - 1)^2$.

$4\sqrt{5}$ $2\sqrt{5}$ $0$ $12$

32

Simplify $(3\sqrt{6} - \sqrt{2})^2$.

$56 - 12\sqrt{3}$ $56 - 6\sqrt{12}$ $52 - 12\sqrt{3}$ $56 + 12\sqrt{3}$

33

Simplify $\sqrt{3}(\sqrt{27} - 2\sqrt{15})$.

$9 - 6\sqrt{5}$ $9 - 2\sqrt{45}$ $9 - 6\sqrt{15}$ $9 - 18\sqrt{5}$

34

Expand and simplify: $(4\sqrt{2} - 3)(5\sqrt{2} + 1)$.

$37 - 11\sqrt{2}$ $37 - 19\sqrt{2}$ $43 - 11\sqrt{2}$ $37 - 9\sqrt{2}$

35

Evaluate $(\sqrt{18} + \sqrt{2})(\sqrt{8} - \sqrt{2})$.

$8$ $4$ $12$ $24$

36

Simplify the expression: $\frac{(2 + \sqrt{3})^2}{2} + \frac{5}{2}$.

$6 + 2\sqrt{3}$ $6 + 4\sqrt{3}$ $4 + 2\sqrt{3}$ $12 + 4\sqrt{3}$

37

Expand and simplify: $(4\sqrt{x} - 2\sqrt{y})(3\sqrt{x} + \sqrt{y})$. Assume $x>0, y>0$.

$12x - 2y - 2\sqrt{xy}$ $12x + 2y - 2\sqrt{xy}$ $12x - 2y - 10\sqrt{xy}$ $12\sqrt{x} - 2\sqrt{y} - 2\sqrt{xy}$

38

If $k = \sqrt{7}$, simplify the expression $(k - 2)(k + 2) + 5$.

$8$ $16$ $5$ $12$

39

Determine the value of $(3 - \sqrt{5})^2 - (3 - \sqrt{5})(3 + \sqrt{5})$.

$10 - 6\sqrt{5}$ $18 - 6\sqrt{5}$ $14 - 6\sqrt{5}$ $6 - 6\sqrt{5}$

40

Given $P = \sqrt{2}(\sqrt{8} + 1)$, find the value of $P^2$.

$18 + 8\sqrt{2}$ $16 + 2\sqrt{16}$ $18 + 4\sqrt{2}$ $16 + 8\sqrt{2}$

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