Question

sec 4.3 mixed and entire radicals
ingrid
section 4.3: mixed and entire radicals

	entire radical	mixed radical
example 1	$sqrt{24}$	$2sqrt{6}$
example 2	$sqrt3{24}$	$2sqrt3{3}$

how???

think about this.....
$sqrt{36} = $
$sqrt3{64} = $

multiplication property of radicals
$sqrtn{ab} = sqrtn{a} \times sqrtn{b}$
where n is a natural number and a and b are real numbers

so....how is $sqrt{24}=2sqrt{6}$ and $sqrt3{24}=2sqrt3{3}$ ?

Explanation:

Step1: Factor 24 for square root

Find largest perfect square factor of 24: $24 = 4 \times 6$

Step2: Apply radical multiplication rule

$\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6}$

Step3: Simplify perfect square root

$\sqrt{4} = 2$, so $\sqrt{24} = 2\sqrt{6}$
---

Step4: Factor 24 for cube root

Find largest perfect cube factor of 24: $24 = 8 \times 3$

Step5: Apply radical multiplication rule

$\sqrt[3]{24} = \sqrt[3]{8 \times 3} = \sqrt[3]{8} \times \sqrt[3]{3}$

Step6: Simplify perfect cube root

$\sqrt[3]{8} = 2$, so $\sqrt[3]{24} = 2\sqrt[3]{3}$
---

Step7: Simplify $\sqrt{36}$

$\sqrt{36} = \sqrt{4 \times 9} = \sqrt{4} \times \sqrt{9} = 2 \times 3 = 6$

Step8: Simplify $\sqrt[3]{64}$

$\sqrt[3]{64} = 4$ (since $4^3 = 64$)

Answer:

• $\sqrt{24}=2\sqrt{6}$ because $\sqrt{24}=\sqrt{4\times6}=\sqrt{4}\times\sqrt{6}=2\sqrt{6}$
• $\sqrt[3]{24}=2\sqrt[3]{3}$ because $\sqrt[3]{24}=\sqrt[3]{8\times3}=\sqrt[3]{8}\times\sqrt[3]{3}=2\sqrt[3]{3}$
• $\sqrt{36}=6$
• $\sqrt[3]{64}=4$