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Simplify:
Simplify $\text{\hspace{0.17em}}{\left(8x\right)}^{\frac{1}{3}}\left(14{x}^{\frac{6}{5}}\right).$
$28{x}^{\frac{23}{15}}$
Access these online resources for additional instruction and practice with radicals and rational exponents.
What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.
When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.
Where would radicals come in the order of operations? Explain why.
Every number will have two square roots. What is the principal square root?
The principal square root is the nonnegative root of the number.
Can a radical with a negative radicand have a real square root? Why or why not?
For the following exercises, simplify each expression.
$\sqrt{\sqrt{256}}$
$\sqrt{289}-\sqrt{121}$
$\sqrt{1}$
$\sqrt{\frac{27}{64}}$
$\sqrt{800}$
$\sqrt{\frac{8}{50}}$
$\sqrt{192}$
$15\sqrt{5}+7\sqrt{45}$
$\sqrt{\frac{96}{100}}$
$12\sqrt{3}-4\sqrt{75}$
$\sqrt{\frac{405}{324}}$
$\frac{5}{1+\sqrt{3}}$
$\sqrt[4]{16}$
$\sqrt[5]{\frac{\mathrm{-32}}{243}}$
$3\sqrt[3]{\mathrm{-432}}+\sqrt[3]{16}$
For the following exercises, simplify each expression.
$\sqrt{4{y}^{2}}$
${\left(144{p}^{2}{q}^{6}\right)}^{\frac{1}{2}}$
$9\sqrt{3{m}^{2}}+\sqrt{27}$
$\frac{4\sqrt{2n}}{\sqrt{16{n}^{4}}}$
$3\sqrt{44z}+\sqrt{99z}$
$\sqrt{490b{c}^{2}}$
${q}^{\frac{3}{2}}\sqrt{63p}$
$\frac{\sqrt{8}}{1-\sqrt{3x}}$
$\frac{2\sqrt{2}+2\sqrt{6x}}{1\mathrm{-3}x}$
$\sqrt{\frac{20}{121{d}^{4}}}$
${w}^{\frac{3}{2}}\sqrt{32}-{w}^{\frac{3}{2}}\sqrt{50}$
$-w\sqrt{2w}$
$\sqrt{108{x}^{4}}+\sqrt{27{x}^{4}}$
$\frac{\sqrt{12x}}{2+2\sqrt{3}}$
$\frac{3\sqrt{x}-\sqrt{3x}}{2}$
$\sqrt{147{k}^{3}}$
$\sqrt{\frac{42q}{36{q}^{3}}}$
$\sqrt{72c}-2\sqrt{2c}$
$\sqrt[3]{24{x}^{6}}+\sqrt[3]{81{x}^{6}}$
$\sqrt[4]{\frac{162{x}^{6}}{16{x}^{4}}}$
$\frac{3\sqrt[4]{2{x}^{2}}}{2}$
$\sqrt[3]{64y}$
$\sqrt[3]{128{z}^{3}}-\sqrt[3]{\mathrm{-16}{z}^{3}}$
$6z\sqrt[3]{2}$
$\sqrt[5]{\mathrm{1,024}{c}^{10}}$
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating $\text{\hspace{0.17em}}\sqrt{\mathrm{90,000}+\mathrm{160,000}}.\text{\hspace{0.17em}}$ What is the length of the guy wire?
500 feet
A car accelerates at a rate of $\text{\hspace{0.17em}}6-\frac{\sqrt{4}}{\sqrt{t}}{\text{m/s}}^{2}\text{\hspace{0.17em}}$ where t is the time in seconds after the car moves from rest. Simplify the expression.
For the following exercises, simplify each expression.
$\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-{2}^{\frac{1}{2}}$
$\frac{\mathrm{-5}\sqrt{2}\mathrm{-6}}{7}$
$\frac{{4}^{\frac{3}{2}}-{16}^{\frac{3}{2}}}{{8}^{\frac{1}{3}}}$
$\frac{\sqrt{m{n}^{3}}}{{a}^{2}\sqrt{{c}^{\mathrm{-3}}}}\cdot \frac{{a}^{\mathrm{-7}}{n}^{\mathrm{-2}}}{\sqrt{{m}^{2}{c}^{4}}}$
$\frac{\sqrt{mnc}}{{a}^{9}cmn}$
$\frac{a}{a-\sqrt{c}}$
$\frac{x\sqrt{64y}+4\sqrt{y}}{\sqrt{128y}}$
$\frac{2\sqrt{2}x+\sqrt{2}}{4}$
$\left(\frac{\sqrt{250{x}^{2}}}{\sqrt{100{b}^{3}}}\right)\left(\frac{7\sqrt{b}}{\sqrt{125x}}\right)$
$\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$
$\frac{\sqrt{3}}{3}$
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