Question

complete the study guide and fill in any additional notes pertaining to each topic.
hr
7.4 objective: simplify radicals involving addition & subtraction
perform indicated operation. simplify final answer.
$4\sqrt{3}-8\sqrt{3}$
$3\sqrt{54}-5\sqrt{24}$
$\sqrt4{32y^{2}z^{5}} + 3z\sqrt4{2y^{2}z}$
$2\sqrt3{54x^{7}}+2\sqrt{27x^{7}}$
$\sqrt3{\frac{216}{w^{6}}}+\sqrt{\frac{121}{w^{4}}}$
additional notes
7.5 objective: multiply radical expressions.
multiply.
$(\sqrt{5}+\sqrt{6})(\sqrt{2}-4)$
$(\sqrt{2}-\sqrt{12})^{2}$
$(2+\sqrt3{5})(2-\sqrt3{5})$
additional notes
foil first outer inner last
objective: rationalize denominators with one radical term
simplify. rationalize the denominator.
$\frac{2}{\sqrt{15}}$
$-\sqrt{\frac{45}{32}}$
$\sqrt3{\frac{16}{x^{2}}}$
additional notes
objective: rationalize denominators with binomials involving radicals.
simplify. rationalize the denominator.
$\frac{4}{\sqrt{3}+2}$
$\frac{5}{\sqrt{3}-\sqrt{10}}$
$\frac{\sqrt{6}+2}{\sqrt{2}-4}$
additional notes

Explanation:

7.4 Objective: Simplify Radicals involving addition & subtraction

Step1: Combine like radicals

$4\sqrt{3}-8\sqrt{3}=(4-8)\sqrt{3}$

Step2: Calculate coefficient

$=-4\sqrt{3}$

---

Step1: Simplify each radical

$3\sqrt{54}=3\sqrt{9\times6}=9\sqrt{6}$, $5\sqrt{24}=5\sqrt{4\times6}=10\sqrt{6}$

Step2: Subtract simplified radicals

$9\sqrt{6}-10\sqrt{6}=(9-10)\sqrt{6}=-\sqrt{6}$

---

Step1: Simplify first radical

$\sqrt[4]{32y^2z^5}=\sqrt[4]{16z^4\times2y^2z}=2z\sqrt[4]{2y^2z}$

Step2: Combine like radicals

$2z\sqrt[4]{2y^2z}+3z\sqrt[4]{2y^2z}=(2z+3z)\sqrt[4]{2y^2z}=5z\sqrt[4]{2y^2z}$

---

Step1: Simplify each radical

$2\sqrt[3]{54x^7}=2\sqrt[3]{27x^6\times2x}=6x^2\sqrt[3]{2x}$, $2\sqrt{27x^7}=2\sqrt{9x^6\times3x}=6x^3\sqrt{3x}$

Step2: Write final expression

$=6x^2\sqrt[3]{2x}+6x^3\sqrt{3x}$

---

Step1: Simplify each radical

$\sqrt[3]{\frac{216}{w^6}}=\frac{6}{w^2}$, $\sqrt{\frac{121}{w^4}}=\frac{11}{w^2}$

Step2: Add simplified terms

$\frac{6}{w^2}+\frac{11}{w^2}=\frac{6+11}{w^2}=\frac{17}{w^2}$

Step1: Apply FOIL method

$(\sqrt{5}+\sqrt{6})(\sqrt{2}-4)=\sqrt{5}\cdot\sqrt{2}-\sqrt{5}\cdot4+\sqrt{6}\cdot\sqrt{2}-\sqrt{6}\cdot4$

Step2: Simplify each term

$=\sqrt{10}-4\sqrt{5}+\sqrt{12}-4\sqrt{6}=\sqrt{10}-4\sqrt{5}+2\sqrt{3}-4\sqrt{6}$

---

Step1: Expand square of binomial

$(\sqrt{2}-\sqrt{12})^2=(\sqrt{2})^2-2\cdot\sqrt{2}\cdot\sqrt{12}+(\sqrt{12})^2$

Step2: Simplify each term

$=2-2\sqrt{24}+12=14-4\sqrt{6}$

---

Step1: Use difference of squares

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

Step2: Calculate final value

$=4-\sqrt[3]{25}$

Step1: Rationalize denominator

$\frac{2}{\sqrt{15}}=\frac{2\cdot\sqrt{15}}{\sqrt{15}\cdot\sqrt{15}}$

Step2: Simplify fraction

$=\frac{2\sqrt{15}}{15}$

---

Step1: Simplify radical first

$-\sqrt{\frac{45}{32}}=-\frac{\sqrt{9\times5}}{\sqrt{16\times2}}=-\frac{3\sqrt{5}}{4\sqrt{2}}$

Step2: Rationalize denominator

$=-\frac{3\sqrt{5}\cdot\sqrt{2}}{4\sqrt{2}\cdot\sqrt{2}}=-\frac{3\sqrt{10}}{8}$

---

Step1: Rewrite radical as fraction

$\sqrt[3]{\frac{16}{x^2}}=\frac{\sqrt[3]{16}}{\sqrt[3]{x^2}}$

Step2: Rationalize denominator

$=\frac{\sqrt[3]{16}\cdot\sqrt[3]{x}}{\sqrt[3]{x^2}\cdot\sqrt[3]{x}}=\frac{\sqrt[3]{16x}}{x}=\frac{2\sqrt[3]{2x}}{x}$

Answer:

$4\sqrt{3}-8\sqrt{3}=-4\sqrt{3}$
$3\sqrt{54}-5\sqrt{24}=-\sqrt{6}$
$\sqrt[4]{32y^2z^5}+3z\sqrt[4]{2y^2z}=5z\sqrt[4]{2y^2z}$
$2\sqrt[3]{54x^7}+2\sqrt{27x^7}=6x^2\sqrt[3]{2x}+6x^3\sqrt{3x}$
$\sqrt[3]{\frac{216}{w^6}}+\sqrt{\frac{121}{w^4}}=\frac{17}{w^2}$

---

7.5 Objective: Multiply Radical Expressions