Question

example 1 add and subtract expressions with like radicands
simplify $3\sqrt{7} + 7\sqrt{7}$.
$3\sqrt{7} + 7\sqrt{7} = (\\_\\_\\_\\_\\_)\sqrt{7}$
$= \\_\\_\\_\\_\\_$

example 2 add and subtract expressions with unlike radicands
simplify $6\sqrt3{40} - 2\sqrt3{135}$.
$6\sqrt3{40} - 2\sqrt3{135}$
$= 6(\\_\\_\\_\\_\\_ \cdot \sqrt3{5}) - 2(\\_\\_\\_\\_\\_ \cdot \sqrt3{5})$ product property of radicals
$= 6(\\_\\_\sqrt3{5}) - 2(\\_\\_\sqrt3{5})$ simplify.
$= \\_\\_\\_\sqrt3{5} - \\_\\_\\_\sqrt3{5}$ multiply.
$= (12 - 6)\sqrt3{5}$ distributive property
$= \\_\\_\\_\\_\\_$ simplify.

check
simplify $-11\sqrt{50} +2\sqrt{32}$.
a. $-63\sqrt{2}$
b. $-47\sqrt{2}$
c. $8\sqrt{2} - 11\sqrt{50}$
d. $-9\sqrt{2}$
lesson 7-6 · operations

Explanation:

Step1: Combine like radicals (Example1)

$3\sqrt{7} + 7\sqrt{7} = (3+7)\sqrt{7}$

Step2: Compute sum (Example1)

$= 10\sqrt{7}$

Step3: Factor radicands (Example2)

$6\sqrt[3]{40} - 2\sqrt[3]{135} = 6(\sqrt[3]{8} \cdot \sqrt[3]{5}) - 2(\sqrt[3]{27} \cdot \sqrt[3]{5})$

Step4: Simplify cube roots (Example2)

$= 6(2\sqrt[3]{5}) - 2(3\sqrt[3]{5})$

Step5: Multiply coefficients (Example2)

$= 12\sqrt[3]{5} - 6\sqrt[3]{5}$

Step6: Combine like radicals (Example2)

$= (12 - 6)\sqrt[3]{5}$

Step7: Compute difference (Example2)

$= 6\sqrt[3]{5}$

Step8: Factor radicands (Check)

$-11\sqrt{50} + 2\sqrt{32} = -11\sqrt{25 \cdot 2} + 2\sqrt{16 \cdot 2}$

Step9: Simplify square roots (Check)

$= -11(5\sqrt{2}) + 2(4\sqrt{2})$

Step10: Multiply coefficients (Check)

$= -55\sqrt{2} + 8\sqrt{2}$

Step11: Combine like radicals (Check)

$= (-55+8)\sqrt{2} = -47\sqrt{2}$

Answer:

Example 1:

Blanks: $3+7$; $10\sqrt{7}$
Final simplified form: $10\sqrt{7}$

Example 2:

Blanks (in order): $\sqrt[3]{8}$; $\sqrt[3]{27}$; $2$; $3$; $12$; $6$; $6\sqrt[3]{5}$
Final simplified form: $6\sqrt[3]{5}$

Check:

B. $-47\sqrt{2}$