Multiplying-Radical-Expressions Worksheet | PDF - Free Printable

Here are the step-by-step solutions for the problems on the worksheet.

1) $\sqrt{5} \times \sqrt{5}$
Multiply the numbers inside: $5 \times 5 = 25$.
$\sqrt{25} = 5$.
Answer: 5

2) $\sqrt{5} \times \sqrt{10}$
Multiply inside: $\sqrt{50}$.
Simplify $\sqrt{50}$ by finding a perfect square factor ($25 \times 2$).
$\sqrt{25} \times \sqrt{2} = 5\sqrt{2}$.
Answer: $5\sqrt{2}$

3) $\sqrt{2} \times \sqrt{18}$
Multiply inside: $\sqrt{36}$.
$\sqrt{36} = 6$.
Answer: 6

4) $\sqrt{14} \times \sqrt{21}$
Multiply inside: $\sqrt{294}$.
Find factors of 294 to simplify. $294 = 49 \times 6$.
$\sqrt{49} \times \sqrt{6} = 7\sqrt{6}$.
Answer: $7\sqrt{6}$

5) $\sqrt{5} \times -4\sqrt{20}$
Multiply coefficients and radicals: $-4 \times \sqrt{100}$.
$\sqrt{100} = 10$.
$-4 \times 10 = -40$.
Answer: -40

6) $3\sqrt{12} \times \sqrt{6}$
Multiply coefficients and radicals: $3 \times \sqrt{72}$.
Simplify $\sqrt{72}$ ($36 \times 2$). $\sqrt{36} = 6$.
$3 \times 6\sqrt{2} = 18\sqrt{2}$.
Answer: $18\sqrt{2}$

7) $5\sqrt{42} \times \sqrt{3}$
Multiply inside: $5\sqrt{126}$.
Simplify $\sqrt{126}$ ($9 \times 14$). $\sqrt{9} = 3$.
$5 \times 3\sqrt{14} = 15\sqrt{14}$.
Answer: $15\sqrt{14}$

8) $\sqrt{3} \times -\sqrt{25}$
$\sqrt{25} = 5$.
So, $\sqrt{3} \times -5 = -5\sqrt{3}$.
Answer: $-5\sqrt{3}$

9) $\sqrt{99} \times \sqrt{48}$
Simplify first: $\sqrt{99} = 3\sqrt{11}$ and $\sqrt{48} = 4\sqrt{3}$.
Multiply: $(3 \times 4)(\sqrt{11} \times \sqrt{3}) = 12\sqrt{33}$.
Answer: $12\sqrt{33}$

10) $5\sqrt{45} \times 3\sqrt{176}$
Simplify first: $5\sqrt{45} = 15\sqrt{5}$ and $3\sqrt{176} = 12\sqrt{11}$.
Multiply: $(15 \times 12)(\sqrt{5} \times \sqrt{11}) = 180\sqrt{55}$.
Answer: $180\sqrt{55}$

11) $\sqrt{12}(3 + \sqrt{3})$
Distribute $\sqrt{12}$ (which is $2\sqrt{3}$):
$2\sqrt{3}(3) + 2\sqrt{3}(\sqrt{3})$
$6\sqrt{3} + 2(3) = 6\sqrt{3} + 6$.
Answer: $6\sqrt{3} + 6$

12) $\sqrt{23x^2} \times \sqrt{23x}$
Combine: $\sqrt{529x^3}$.
$\sqrt{529} = 23$. $\sqrt{x^3} = x\sqrt{x}$.
Answer: $23x\sqrt{x}$

13) $-5\sqrt{12} \times -\sqrt{3}$
Negatives cancel out. Simplify $\sqrt{12}$ to $2\sqrt{3}$.
$5(2\sqrt{3}) \times \sqrt{3} = 10\sqrt{3} \times \sqrt{3} = 10(3)$.
Answer: 30

14) $2\sqrt{20x^2} \times \sqrt{5x^2}$
Simplify first term: $2(2x)\sqrt{5} = 4x\sqrt{5}$.
Second term: $x\sqrt{5}$.
Multiply: $(4x)(x)(\sqrt{5})(\sqrt{5}) = 4x^2(5)$.
Answer: $20x^2$

15) $\sqrt{12x^2} \times \sqrt{2x^3}$
Combine: $\sqrt{24x^5}$.
Split into perfect squares: $\sqrt{4x^4 \cdot 6x}$.
Pull out roots: $2x^2\sqrt{6x}$.
Answer: $2x^2\sqrt{6x}$

16) $-12\sqrt{7x} \times \sqrt{5x^3}$
Combine: $-12\sqrt{35x^4}$.
Pull out $x^2$: $-12x^2\sqrt{35}$.
Answer: $-12x^2\sqrt{35}$

17) $-5\sqrt{9x^3} \times 6\sqrt{3x^2}$
Simplify first: $-5(3x)\sqrt{x} = -15x\sqrt{x}$.
Simplify second: $6(x)\sqrt{3} = 6x\sqrt{3}$.
Multiply: $(-15x)(6x)(\sqrt{x}\sqrt{3}) = -90x^2\sqrt{3x}$.
Answer: $-90x^2\sqrt{3x}$

18) $-2\sqrt{12}(3 + \sqrt{12})$
Simplify $\sqrt{12}$ to $2\sqrt{3}$. Term becomes $-4\sqrt{3}$.
Distribute $-4\sqrt{3}(3 + 2\sqrt{3})$.
$-12\sqrt{3} - 8(3) = -12\sqrt{3} - 24$.
Answer: $-12\sqrt{3} - 24$

19) $\sqrt{18x}(4 - \sqrt{6x})$
Simplify $\sqrt{18x}$ to $3\sqrt{2x}$.
Distribute $3\sqrt{2x}(4 - \sqrt{6x})$.
$12\sqrt{2x} - 3\sqrt{12x^2}$.
Simplify second part: $3(2x)\sqrt{3} = 6x\sqrt{3}$.
Answer: $12\sqrt{2x} - 6x\sqrt{3}$

20) $\sqrt{3x}(6\sqrt{x^3} + \sqrt{27})$
Distribute $\sqrt{3x}$.
First part: $6\sqrt{3x^4} = 6x^2\sqrt{3}$.
Second part: $\sqrt{81x} = 9\sqrt{x}$.
Answer: $6x^2\sqrt{3} + 9\sqrt{x}$

21) $\sqrt{15r}(5 + \sqrt{5})$
Distribute $\sqrt{15r}$.
$5\sqrt{15r} + \sqrt{75r}$.
Simplify $\sqrt{75r}$ to $5\sqrt{3r}$.
Answer: $5\sqrt{15r} + 5\sqrt{3r}$

22) $-5\sqrt{3x} \times 4\sqrt{6x^3}$
Multiply coeffs: $-20$.
Multiply radicals: $\sqrt{18x^4}$.
Simplify radical: $3x^2\sqrt{2}$.
Total: $-20(3x^2\sqrt{2})$.
Answer: $-60x^2\sqrt{2}$

23) $-2\sqrt{18x} \times 4\sqrt{2x}$
Multiply coeffs: $-8$.
Multiply radicals: $\sqrt{36x^2} = 6x$.
Total: $-8(6x)$.
Answer: $-48x$

24) $-3\sqrt{5v^2} (-3\sqrt{15v})$
Multiply coeffs: $(-3)(-3) = 9$.
Multiply radicals: $\sqrt{75v^3}$.
Simplify radical: $\sqrt{25v^2 \cdot 3v} = 5v\sqrt{3v}$.
Total: $9(5v\sqrt{3v})$.
Answer: $45v\sqrt{3v}$

25) $(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})$
Difference of squares pattern: $a^2 - b^2$.
$(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3$.
Answer: 2

26) $(-4\sqrt{6} + 2)(\sqrt{6} - 5)$
FOIL method:
First: $-4(6) = -24$.
Outer: $20\sqrt{6}$.
Inner: $2\sqrt{6}$.
Last: $-10$.
Combine: $-34 + 22\sqrt{6}$.
Answer: $-34 + 22\sqrt{6}$

27) $(2 - 2\sqrt{3})(-2 + \sqrt{3})$
FOIL method:
First: $-4$.
Outer: $2\sqrt{3}$.
Inner: $4\sqrt{3}$.
Last: $-2(3) = -6$.
Combine: $-10 + 6\sqrt{3}$.
Answer: $-10 + 6\sqrt{3}$

28) $(11 - 4\sqrt{5})(6 - \sqrt{5})$
FOIL method:
First: $66$.
Outer: $-11\sqrt{5}$.
Inner: $-24\sqrt{5}$.
Last: $+4(5) = +20$.
Combine: $86 - 35\sqrt{5}$.
Answer: $86 - 35\sqrt{5}$

29) $(-2 - \sqrt{3x})(3 + \sqrt{3x})$
FOIL method:
First: $-6$.
Outer: $-2\sqrt{3x}$.
Inner: $-3\sqrt{3x}$.
Last: $-3x$.
Combine: $-6 - 5\sqrt{3x} - 3x$.
Answer: $-6 - 5\sqrt{3x} - 3x$

30) $(-2 + 3\sqrt{2r})(-2 + \sqrt{2r})$
FOIL method:
First: $4$.
Outer: $-2\sqrt{2r}$.
Inner: $-6\sqrt{2r}$.
Last: $3(2r) = 6r$.
Combine: $4 - 8\sqrt{2r} + 6r$.
Answer: $4 - 8\sqrt{2r} + 6r$

31) $(-4\sqrt{2n} + 2)(-2\sqrt{2} - 4)$
FOIL method:
First: $8\sqrt{4n} = 16\sqrt{n}$.
Outer: $16\sqrt{2n}$.
Inner: $-4\sqrt{2}$.
Last: $-8$.
Answer: $16\sqrt{n} + 16\sqrt{2n} - 4\sqrt{2} - 8$

32) $(-1 + 2\sqrt{3})(2 - 3\sqrt{3x})$
FOIL method:
First: $-2$.
Outer: $3\sqrt{3x}$.
Inner: $4\sqrt{3}$.
Last: $-6\sqrt{9x} = -18\sqrt{x}$.
Answer: $-2 + 3\sqrt{3x} + 4\sqrt{3} - 18\sqrt{x}$