Homework for 14 October
Each part has its solution folded. Do the exercise in your notebook and, when you have it, click on the statement to unfold it and check. Only the parts marked in class.
32Carry out the following operations and simplify the result
d) $\sqrt{5^{7}}$
$$\sqrt{5^{7}}=\sqrt{5^{6}\cdot 5}=5^{3}\sqrt{5}=125\sqrt{5}$$
f) $\sqrt[6]{2^{8}}$
$$\sqrt[6]{2^{8}}=\sqrt[3]{2^{4}}=\sqrt[3]{2^{3}\cdot 2}=2\cdot\sqrt[3]{2}$$
33Write the following powers in radical form
a) $3^{2/3}$
$$3^{2/3}=\sqrt[3]{3^{2}}$$
c) $\left(\dfrac{4}{3}\right)^{7/12}$
$$\left(\dfrac{4}{3}\right)^{7/12}=\sqrt[12]{\left(\dfrac{4}{3}\right)^{7}}$$
f) $\left(-\dfrac{1}{7}\right)^{3/5}$
$$\left(-\dfrac{1}{7}\right)^{3/5}=\sqrt[5]{\left(-\dfrac{1}{7}\right)^{3}}$$
34Take out all possible factors, numerical and literal, from these radicals
h) $\sqrt[4]{x^{5}y^{8}z}$
From each letter we take out as many factors as fit: $x^5=x^4\cdot x$ and $y^8=(y^2)^4$.
$$\sqrt[4]{x^{5}y^{8}z}=x\,y^{2}\,\sqrt[4]{x\,z}$$
i) $\sqrt[3]{54x^{3}y^{4}z^{12}}$
$54=2\cdot 3^{3}$.
$$\sqrt[3]{54x^{3}y^{4}z^{12}}=\sqrt[3]{2\cdot 3^{3}\cdot x^{3}\cdot y^{3}\cdot y\cdot z^{12}}=3\,x\,y\,z^{4}\,\sqrt[3]{2y}$$
j) $\sqrt[5]{972x^{7}y^{5}z^{10}}$
$972=2^{2}\cdot 3^{5}$.
$$\sqrt[5]{972x^{7}y^{5}z^{10}}=\sqrt[5]{2^{2}\cdot 3^{5}\cdot x^{5}\cdot x^{2}\cdot y^{5}\cdot z^{10}}=3\,x\,y\,z^{2}\,\sqrt[5]{4x^{2}}$$
35Work out the result of the following operations
c) $\sqrt{12}-3\sqrt{27}+5\sqrt{3}+\sqrt{75}$
$12=2^2\cdot 3,\ 27=3^3,\ 75=5^2\cdot 3$.
$$2\sqrt{3}-3\cdot 3\sqrt{3}+5\sqrt{3}+5\sqrt{3}=(2-9+5+5)\sqrt{3}=3\sqrt{3}$$
d) $\sqrt[3]{40}+\sqrt{50}-3\sqrt{28}+2\sqrt{2}-2\sqrt[3]{5}+2\sqrt[3]{125}$
$40=2^3\cdot 5,\ 50=2\cdot 5^2,\ 28=2^2\cdot 7,\ \sqrt[3]{125}=5$.
$$2\sqrt[3]{5}+5\sqrt{2}-6\sqrt{7}+2\sqrt{2}-2\sqrt[3]{5}+10$$
The $\sqrt[3]{5}$ terms cancel and the $\sqrt{2}$ terms add up; $\sqrt{7}$ cannot be combined.
$$10+7\sqrt{2}-6\sqrt{7}$$
36Carry out the proposed operations
All with positive bases.
e) $\sqrt[3]{5^{2}}\cdot\sqrt[6]{5}\cdot\sqrt{5}$
With fractional exponents:
$$5^{2/3}\cdot 5^{1/6}\cdot 5^{1/2}=5^{(4+1+3)/6}=5^{8/6}=5^{4/3}=\sqrt[3]{5^{4}}=5\cdot\sqrt[3]{5}$$
f) $\sqrt{\left(\dfrac{1}{5}\right)^{3}}:\sqrt[3]{\left(\dfrac{1}{5}\right)^{5}}\cdot\sqrt[4]{\dfrac{1}{5}}$
$$\left(\dfrac{1}{5}\right)^{3/2-5/3+1/4}=\left(\dfrac{1}{5}\right)^{(18-20+3)/12}=\left(\dfrac{1}{5}\right)^{1/12}=\sqrt[12]{\dfrac{1}{5}}$$
h) $\sqrt{\sqrt[4]{2}}\cdot\left(\sqrt[5]{2}\right)^{2}$
$$2^{1/8}\cdot 2^{2/5}=2^{(5+16)/40}=2^{21/40}=\sqrt[40]{2^{21}}$$
k) $\sqrt{\sqrt[3]{a^{20}}}:\sqrt[6]{a^{8}}$
$$a^{20/6}:a^{8/6}=a^{12/6}=a^{2}$$
l) $\sqrt[4]{a^{7}b^{4}}:\sqrt{\sqrt{a^{3}b^{4}}}$
$\sqrt{\sqrt{a^3b^4}}$ is a fourth root. With $a,b>0$:
$$\sqrt[4]{a^{7}b^{4}}:\sqrt[4]{a^{3}b^{4}}=\sqrt[4]{\dfrac{a^{7}b^{4}}{a^{3}b^{4}}}=\sqrt[4]{a^{4}}=a$$