smc-4214-30-Roots

Indices, SM-Bank 079

Evaluate \(3\times\sqrt{81}+\sqrt[3]{27}\times 2\). (2 marks)

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\(33\)

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\(3\times \sqrt{81}+\sqrt[3]{27}\times 2\)	\(=3\times\sqrt{9\times 9}+\sqrt[3]{3\times 3\times 3}\times 2\)
	\(=3\times 9+3\times 2\)
	\(=33\)

Indices, SM-Bank 078

Evaluate \(3\times\sqrt[3]{64}-\sqrt{100}\). (2 marks)

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\(2\)

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\(3\times \sqrt[3]{64}-\sqrt{100}\)	\(=3\times\sqrt[3]{4\times 4\times 4}-\sqrt{10\times 10}\)
	\(=3\times 4-10\)
	\(=2\)

Indices, SM-Bank 077

Evaluate \(\sqrt[3]{27}-\sqrt{25}\). (2 marks)

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\(-2\)

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\(\sqrt[3]{27}-\sqrt{25}\)	\(=\sqrt[3]{3\times 3\times 3}-\sqrt{5\times 5}\)
	\(=3-5\)
	\(=-2\)

Indices, SM-Bank 076

Evaluate \(\sqrt{9}+\sqrt[3]{8}\). (2 marks)

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\(5\)

Show Worked Solution

\(\sqrt{9}+\sqrt[3]{8}\)	\(=\sqrt{3\times 3}+\sqrt[3]{2\times 2\times 2}\)
	\(=3+2\)
	\(=5\)

Indices, SM-Bank 075

Find the missing whole number that makes the following number sentence correct. (2 marks)

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Show Worked Solution

\(\text{Using trial and error method:}\)

\(\text{Test 1st square number}\ \rightarrow 1\)

\(\text{LHS:}\)	\(\ 3\times 1^2+(\sqrt{25}-1)=3\times 1+(5-1)\)
	\(=7\)
\(\text{RHS:}\)	\(\ \sqrt{1}\times 9\times 3-5=1\times 27-5\)
	\(=23\)

\(\text{LHS }\ne\text{ RHS}\)

\(\text{Test 2nd square number}\ \rightarrow 4\)

\(\text{LHS:}\)	\(\ 3\times 4^2+(\sqrt{25}-4)=3\times 16+(5-4)\)
	\(=49\)
\(\text{RHS:}\)	\(\ \sqrt{4}\times 9\times 3-5=2\times 27-5\)
	\(=49\)

\(\text{LHS }=\text{ RHS}\)

\(\therefore \ \)

Indices, SM-Bank 074 MC

Between which two number does \(\sqrt{70}\) lie?

\(5\ \text{and }6\)
\(6\ \text{and }7\)
\(7\ \text{and }8\)
\(8\ \text{and }9\)

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\(D\)

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\(\text{Consider Option D:}\)	\(\rightarrow\ \sqrt{64}=8\)
	\(\rightarrow\ \sqrt{81}=9\)

\(\therefore\ \sqrt{70}\ \text{lies between }\sqrt{64} \text{ and }\sqrt{81}\)

\(\therefore\ \sqrt{70}\ \text{lies between }8 \text{ and }9\)

\(\Rightarrow D\)

Indices, SM-Bank 073 MC

Between which two number does \(\sqrt{30}\) lie?

\(3\ \text{and }4\)
\(4\ \text{and }5\)
\(5\ \text{and }6\)
\(6\ \text{and }7\)

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\(C\)

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\(\text{Consider Option C:}\)	\(\rightarrow\ \sqrt{25}=5\)
	\(\rightarrow\ \sqrt{36}=6\)

\(\therefore\ \sqrt{30}\ \text{lies between }\sqrt{25} \text{ and }\sqrt{36}\)

\(\therefore\ \sqrt{30}\ \text{lies between }5 \text{ and }6\)

\(\Rightarrow C\)

Indices, SM-Bank 072

Show that \(\sqrt{9\times 4}=\sqrt{9}\times \sqrt{4}\). (2 marks)

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\(\text{See worked solution}\)

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\(\text{LHS}:\sqrt{9\times 4}\)	\(=\sqrt{ 3\times 3\times 2\times 2}\)
	\(=\sqrt{ 6\times 6}\)
	\(=6\)

\(\text{RHS}:\sqrt{9}\times \sqrt{4}\)	\(=3\times 2\)
	\(=6\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{9\times 4}=\sqrt{9}\times \sqrt{4}\)

Indices, SM-Bank 071

Show that \(\sqrt{25\times 16}=\sqrt{25}\times \sqrt{16}\). (2 marks)

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\(\text{See worked solution}\)

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\(\text{LHS}:\sqrt{25\times 16}\)	\(=\sqrt{ 5\times 5\times 4\times 4}\)
	\(=\sqrt{ 20\times 20}\)
	\(=20\)

\(\text{RHS}:\sqrt{25}\times \sqrt{16}\)	\(=5\times 4\)
	\(=20\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{25\times 16}=\sqrt{25}\times \sqrt{16}\)

Indices, SM-Bank 070

Show that \(\sqrt{225}=\sqrt{25}\times \sqrt{9}\). (2 marks)

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\(\text{See worked solution}\)

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\(\text{LHS}:\sqrt{225}\)	\(=\sqrt{ 5\times 45}\)
	\(=\sqrt{ 5\times 5\times 9}\)
	\(=\sqrt{ 5\times 5\times 3\times 3}\)
	\(=\sqrt{ 15\times 15}\)
	\(=15\)

\(\text{RHS}:\sqrt{25}\times \sqrt{9}\)	\(=5\times 3\)
	\(=15\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{225}=\sqrt{25}\times \sqrt{9}\)

Indices, SM-Bank 069

Show that \(\sqrt{144}=\sqrt{36}\times \sqrt{4}\). (2 marks)

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\(\text{See worked solution}\)

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\(\text{LHS}:\sqrt{144}\)	\(=\sqrt{ 12\times 12}\)
	\(=12\)

\(\text{RHS}:\sqrt{36}\times \sqrt{4}\)	\(=6\times 2\)
	\(=12\)

\(\therefore\ \text{LHS}\ =\ \text{RHS}\)

\(\therefore\ \sqrt{144}=\sqrt{36}\times \sqrt{4}\)

Indices, SM-Bank 068 MC

Given that \(12^2=144\), then \(\sqrt{144}=\) ?

\(288\)
\(72\)
\(12\)
\(6\)

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\(C\)

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\(144=12\times 12\ \ \text{(Given)}\)

\(\therefore \sqrt{144}\)	\(=\sqrt{ 12\times 12}\)
	\(=12\)

\(\Rightarrow C\)

Indices, SM-Bank 067 MC

Given that \(17^2=289\), then \(\sqrt{289}=\) ?

\(8.5\)
\(13.5\)
\(17\)
\(578\)

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\(C\)

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\(289=17\times 17\ \ \text{(Given)}\)

\(\therefore \sqrt{289}\)	\(=\sqrt{ 17\times 17}\)
	\(=17\)

\(\Rightarrow C\)

Indices, SM-Bank 066 MC

Given that \(21^2=441\), then \(\sqrt{441}=\) ?

\(21\)
\(42\)
\(420\)
\(194\ 481\)

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\(A\)

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\(441=21\times 21\ \ \text{(Given)}\)

\(\therefore \sqrt{441}\)	\(=\sqrt{ 21\times 21}\)
	\(=21\)

\(\Rightarrow A\)

Indices, SM-Bank 065 MC

Given that \(4^3=64\), then \(\sqrt[3]{64}=\) ?

\(2\)
\(4\)
\(8\)
\(21.3\)

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\(B\)

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\(64=4\times 4\times 4\ \ \text{(Given)}\)

\(\therefore \sqrt[3]{64}\)	\(=\sqrt[3]{ 4\times 4\times 4}\)
	\(=4\)

\(\Rightarrow B\)

Indices, SM-Bank 064 MC

Given that \(8^3=512\), then \(\sqrt[3]{512}=\) ?

\(8\)
\(23\)
\(128\)
\(171\)

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\(A\)

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\(512=8\times 8\times 8\ \ \text{(Given)}\)

\(\therefore \sqrt[3]{512}\)	\(=\sqrt[3]{ 8\times 8\times 8}\)
	\(=8\)

\(\Rightarrow A\)

Indices, SM-Bank 063 MC

Given that \(5^3=125\), then \(\sqrt[3]{125}=\) ?

\(62.5\)
\(41.7\)
\(11.2\)
\(5\)

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\(D\)

Show Worked Solution

\(125=5\times 5\times 5\ \ \text{(Given)}\)

\(\therefore \sqrt[3]{125}\)	\(=\sqrt[3]{ 5\times 5\times 5}\)
	\(=5\)

\(\Rightarrow D\)

Indices, SM-Bank 062

Write 900 as a product of its prime factors. (2 marks)

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Hence find \(\sqrt{900}\). (2 marks)

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a. \(2\times 2\times 3\times 3\times 5\times 5\)

b. \(30\)

Show Worked Solution

a.	\(900\)	\(=9\times 100\)
		\(=3\times 3\times 10\times 10\)
		\(=3\times 3\times 2\times 5\times 2\times 5\)
		\(=2\times 2\times 3\times 3\times 5\times 5\)

b.	\(\sqrt{900}\)	\(=\sqrt{2\times 2\times 3\times 3\times 5\times 5}\)
		\(=\sqrt{(2\times 3\times 5)\times (2\times 3\times 5)}\)
		\(=\sqrt{30\times 30}\)
		\(=30\)

Indices, SM-Bank 061

Write 1024 as a product of its prime factors. (2 marks)

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Hence find \(\sqrt{1024}\). (2 marks)

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a. \(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\)

b. \(32\)

Show Worked Solution

a.	\(1024\)	\(=2\times 512\)
		\(=2\times 2\times 256\)
		\(=2\times 2\times 2\times 128\)
		\(=2\times 2\times 2\times 2\times 64\)
		\(=2\times 2\times 2\times 2\times 8\times 8\)
		\(=2\times 2\times 2\times 2\times 2\times 4\times 2\times 4\)
		\(=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\)

b.	\(\sqrt{1024}\)	\(=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}\)
		\(=\sqrt{(2\times 2\times 2\times 2\times 2)\times (2\times 2\times 2\times 2\times 2)}\)
		\(=\sqrt{32\times 32}\)
		\(=32\)

Indices, SM-Bank 060

Write 324 as a product of its prime factors. (2 marks)

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Hence find \(\sqrt{324}\). (2 marks)

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a. \(2\times 2\times 3\times 3\times 3\times 3\)

b. \(18\)

Show Worked Solution

a.	\(324\)	\(=2\times 162\)
		\(=2\times 2\times 81\)
		\(=2\times 2\times 9\times 9\)
		\(=2\times 2\times 3\times 3\times 3\times 3\)

b.	\(\sqrt{324}\)	\(=\sqrt{2\times 2\times 3\times 3\times 3\times 3}\)
		\(=\sqrt{(2\times 3\times 3)\times (2\times 3\times 3)}\)
		\(=\sqrt{18\times 18}\)
		\(=18\)

Indices, SM-Bank 059

Write 256 as a product of its prime factors. (2 marks)

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Hence find \(\sqrt{256}\). (2 marks)

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a. \(2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\)

b. \(16\)

Show Worked Solution

a.	\(256\)	\(=2\times 128\)
		\(=2\times 2\times 64\)
		\(=2\times 2\times 2\times 32\)
		\(=2\times 2\times 2\times 2\times 16\)
		\(=2\times 2\times 2\times 2\times 2\times 8\)
		\(=2\times 2\times 2\times 2\times 2\times 2\times 4\)
		\(=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\)

b.	\(\sqrt{256}\)	\(=\sqrt{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}\)
		\(=\sqrt{(2\times 2\times 2\times 2)\times (2\times 2\times 2\times 2)}\)
		\(=\sqrt{16\times 16}\)
		\(=16\)

Indices, SM-Bank 058

Write 216 as a product of its prime factors. (2 marks)

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Hence find \(\sqrt[3]{216}\). (2 marks)

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a. \(2\times 2\times 2\times 3\times 3\times 3\)

b. \(6\)

Show Worked Solution

a.	\(216\)	\(=2\times 108\)
		\(=2\times 2\times 54\)
		\(=2\times 2\times 2\times 27\)
		\(=2\times 2\times 2\times 3\times 9\)
		\(=2\times 2\times 2\times 3\times 3\times 3\)

b.	\(\sqrt[3]{216}\)	\(=\sqrt[3]{2\times 2\times 2\times 3\times 3\times 3}\)
		\(=\sqrt[3]{(2\times 3)\times (2\times 3)\times (2\times 3)}\)
		\(=\sqrt[3]{6\times 6\times 6}\)
		\(=6\)

Indices, SM-Bank 057

Write 8 as a product of its prime factors. (2 marks)

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Hence find \(\sqrt[3]{8}\). (1 mark)

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a. \(2\times 2\times 2\)

b. \(2\)

Show Worked Solution

a.	\(8\)	\(=2\times 4\)
		\(=2\times 2\times 2\)

b.	\(\sqrt[3]{8}\)	\(=\sqrt[3]{2\times 2\times 2}\)
		\(=2\)