Contents of Powers and Radicals

Powers and Radicals

To raise complex numbers to powers, use common notation for powers and apply Evaluate.

$\blacktriangleright$ Evaluate

i² = -1 4pt

$\left(\vphantom{ 3+2i}\right.$ 3 + 2i $\left.\vphantom{ 3+2i}\right)^{{4}}_{}$ = -119 + 120i 4pt

$\left(\vphantom{ 3+2i}\right.$ 3 + 2i $\left.\vphantom{ 3+2i}\right)^{{-4}}_{}$ = - ${\frac{{119}}{{28561}}}$ - ${\frac{{120}}{{28561}}}$ i 4pt

$\left(\vphantom{ \dfrac{2}{5}-\dfrac{3}{4}i}\right.$ ${\dfrac{{2}}{{5}}}$ - ${\dfrac{{3}}{{4}}}$ i $\left.\vphantom{ \dfrac{2}{5}-\dfrac{3}{4}i}\right)^{{5}}_{}$ = ${\frac{{113221}}{{% 400000}}}$ + ${\frac{{43737}}{{128000}}}$ i 4pt

$\left(\vphantom{ 0.4-.75i}\right.$ 0.4 - .75i $\left.\vphantom{ 0.4-.75i}\right)^{{5}}_{}$ = . 28305 + . 3417i 4pt

$\sqrt{{2.34-i}}$ = 1. 5628 - . 31994i 4pt

$\left(\vphantom{ 2.5+0.5i}\right.$ 2.5 + 0.5i $\left.\vphantom{ 2.5+0.5i}\right)^{{\frac{4}{5}}}_{}$ = 2. 088 + . 3325i pt

$\left(\vphantom{ a+bi}\right.$ a + bi $\left.\vphantom{ a+bi}\right)^{{-1}}_{}$ = ${\frac{{a}}{{a^{2}+b^{2}}}}$ - i ${\frac{{b}}{{% a^{2}+b^{2}}}}$ 4pt

$\left(\vphantom{ 0.16-3i}\right.$ 0.16 - 3i $\left.\vphantom{ 0.16-3i}\right)^{{-1}}_{}$ = 1. 7727×10^-2 + . 33239i 4pt

$\left(\vphantom{ 8i}\right.$ 8i $\left.\vphantom{ 8i}\right)^{{\frac{1}{3}}}_{}$ = ${\frac{{1}}{{2}}}$ $\sqrt[3]{{8}}$ $\sqrt{{3}}$ + ${\frac{{1}}{{2}}}$ i $\sqrt[3]{{8}}$ 4pt

$\sqrt{{2+3i}}$ = ${\frac{{1}}{{2}}}$ $\sqrt{{\left( 4+2\sqrt{13}\right) }}$ + ${\frac{{% 1}}{{2}}}$ i $\sqrt{{\left( -4+2\sqrt{13}\right) }}$

$\sqrt[3]{{i}}$ = ${\frac{{1}}{{2}}}$ $\sqrt{{3}}$ + ${\frac{{1}}{{2}}}$ i

$\sqrt{{1+i}}$ = ${\frac{{1}}{{2}}}$ $\sqrt{{\left( 2+2\sqrt{2}\right) }}$ + ${\frac{{1}}{{2}}}$ i $\sqrt{{% \left( -2+2\sqrt{2}\right) }}$

Note that Evaluate returns a different answer for $\left(\vphantom{ \frac{2}{5}-\frac{3}{4}i}\right.$ ${\frac{{2}}{{5}}}$ - ${\frac{{3}}{{4}}}$ i $\left.\vphantom{ \frac{2}{5}-\frac{3}{4}i}\right)^{{5}}_{}$ and $\left(\vphantom{ 0.4-.75i}\right.$ 0.4 - .75i $\left.\vphantom{ 0.4-.75i}\right)^{{5}}_{}$ . The fraction displayed for $\left(\vphantom{ \frac{2}{5}-\frac{3}{4}i}\right.$ ${\frac{{2}}{{5}}}$ - ${\frac{{3}}{{4}}}$ i $\left.\vphantom{ \frac{2}{5}-\frac{3}{4}i}\right)^{{5}}_{}$ is the exact answer, and the number displayed for $\left(\vphantom{ 0.4-.75i}\right.$ 0.4 - .75i $\left.\vphantom{ 0.4-.75i}\right)^{{5}}_{}$ is the best 5-digit approximation to the exact answer.

For some roots you can obtain the answer in simpler form by applying Simplify after (or in place of) Evaluate.

$\blacktriangleright$ Evaluate

$\left(\vphantom{ 8i}\right.$ 8i $\left.\vphantom{ 8i}\right)^{{\frac{1}{3}}}_{}$ = ${\frac{{1}}{{2}}}$ $\sqrt[3]{{8}}$ $\sqrt{{3}}$ + ${\frac{{1}}{{2}}}$ i $\sqrt[3]{{8}}$

$\blacktriangleright$ Simplify

$\left(\vphantom{ 8i}\right.$ 8i $\left.\vphantom{ 8i}\right)^{{\frac{1}{3}}}_{}$ = $\sqrt{{3}}$ + i

$\blacktriangleright$ Evaluate Numerically

$\left(\vphantom{ 8i}\right.$ 8i $\left.\vphantom{ 8i}\right)^{{\frac{1}{3}}}_{}$ = 1. 7321 + 1.0i

Tip To enter the 3 in $\sqrt[3]{{1+i}}$ , do one of the following.

Click itbpF0.2992in0.2992in0.0692inradical.wmf; place the insertion point inside the radical sign, press the TAB key, and enter the 3 in the small input box that appears;

- or -

Select the radical sign; click itbpF0.3001in0.3001in0.0701inproperty.wmf or choose Edit + Properties; and enter the 3 in the small input box that appears.

You can find the real and imaginary parts of a complex number with the functions $\func$ Re and $\func$ Im. When you enter these functions in mathematics mode, they will automatically turn gray.

$\blacktriangleright$ Evaluate

$\func$ Re $\left(\vphantom{ \dfrac{a+bi}{c+di}}\right.$ ${\dfrac{{a+bi}}{{c+di}}}$ $\left.\vphantom{ \dfrac{a+bi}{c+di}}\right)$ = ${\dfrac{{ac}}{{c^{2}+d^{2}}}}$ + ${\dfrac{{bd}}{{c^{2}+d^{2}}}}$ 6pt = ${\dfrac{{ac+bd}}{{c^{2}+d^{2}}}}$

$\func$ Im $\left(\vphantom{ \dfrac{a+bi}{c+di}}\right.$ ${\dfrac{{a+bi}}{{c+di}}}$ $\left.\vphantom{ \dfrac{a+bi}{c+di}}\right)$ = ${\dfrac{{bc}}{{c^{2}+d^{2}}}}$ - ${\dfrac{{ad}}{{c^{2}+d^{2}}}}$ 6pt = ${\dfrac{{bc-ad}}{{c^{2}+d^{2}}}}$

${\dfrac{{a+bi}}{{c+di}}}$ = ${\dfrac{{ac+bd}}{{c^{2}+d^{2}}}}$ + i $\left(\vphantom{ \dfrac{bc-ad}{% c^{2}+d^{2}}}\right.$ ${\dfrac{{bc-ad}}{{% c^{2}+d^{2}}}}$ $\left.\vphantom{ \dfrac{bc-ad}{% c^{2}+d^{2}}}\right)$ 6pt

$\func$ Re $\left(\vphantom{ \dfrac{3.6+6i}{5-3.25i}}\right.$ ${\dfrac{{3.6+6i}}{{5-3.25i}}}$ $\left.\vphantom{ \dfrac{3.6+6i}{5-3.25i}}\right)$ = - 4.2179×10^-2

$\func$ Im $\left(\vphantom{ \dfrac{3.6+6i}{5-3.25i}}\right.$ ${\dfrac{{3.6+6i}}{{5-3.25i}}}$ $\left.\vphantom{ \dfrac{3.6+6i}{5-3.25i}}\right)$ = 1.1726