# Absolute Value of Power

## Theorem

Let $x$, $y$ be real numbers.

Let $x^y$, $x$ to the power of $y$, be real.

Then:

$\left \vert {x^y} \right \vert = \left \vert {x} \right \vert ^y$

## Proof

If $x = 0$, the theorem clearly holds, by the definition of powers of zero.

Suppose $x \ne 0$.

We use the interpretation of real numbers as wholly real complex numbers.

Likewise we interpret the absolute value of $x$ as the modulus of $x$.

Then $x$ can be expressed in polar form:

$x = r e^{i\theta}$

where $r = \left \vert {x}\right \vert$ and $\theta$ is an argument of $x$.

Then:

 $\displaystyle x$ $=$ $\displaystyle r e^{i\theta}$ $\quad$ $\quad$ $\displaystyle \implies \ \$ $\displaystyle x^y$ $=$ $\displaystyle \left(r{e^{i\theta} }\right)^y$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle r^y e^{i \theta y}$ $\quad$ $\quad$ $\displaystyle \implies \ \$ $\displaystyle \left \vert {x^y} \right \vert$ $=$ $\displaystyle \left \vert {r^y e^{i \theta y} } \right \vert$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \left \vert {r^y} \right \vert \left \vert {e^{i \theta y} } \right \vert$ $\quad$ Modulus of Product $\quad$ $\displaystyle$ $=$ $\displaystyle \left \vert {r^y} \right \vert$ $\quad$ Modulus of Exponential of Imaginary Number is One $\quad$ $\displaystyle$ $=$ $\displaystyle \left \vert {\left \vert {x} \right \vert^y} \right \vert$ $\quad$ by definition of $r$ $\quad$ $\displaystyle$ $=$ $\displaystyle \left \vert {x} \right \vert^y$ $\quad$ as $\left \vert {x} \right \vert^y \ge 0$ $\quad$

$\blacksquare$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9961773753166199, "perplexity": 69.51296699324269}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584431529.98/warc/CC-MAIN-20190123234228-20190124020228-00073.warc.gz"}