index law quesrtion
- Thread starter neelam
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HallsofIvy
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I think Subhotosh Khan means that you can go directly from \(\displaystyle \frac{1}{\sqrt[3]{x^2}}= \frac{64}{9}\) to \(\displaystyle \sqrt[3]{x^2}= \frac{9}{64}\) by inverting both sides: if \(\displaystyle \frac{a}{b}= \frac{c}{d}\) then \(\displaystyle \frac{b}{a}= \frac{d}{c}\).
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I would do it following way
\(\displaystyle \displaystyle x^{-\frac{2}{3}} \ = \ 7\frac{1}{9}\)
\(\displaystyle \displaystyle x^{-\frac{2}{3}} \ = \ \frac{64}{9}\)
\(\displaystyle \displaystyle x^{\frac{2}{3}} \ = \ \left (\frac{3}{8}\right )^2\)
\(\displaystyle \displaystyle \left [ x^{\frac{2}{3}}\right ]^{\frac{3}{2}} \ = \ \left [\left (\frac{3}{8}\right )^2 \right ]^{\frac{3}{2}}\)
\(\displaystyle \displaystyle x \ = \ \left (\frac{3}{8}\right )^3 \)
\(\displaystyle \displaystyle x^{-\frac{2}{3}} \ = \ 7\frac{1}{9}\)
\(\displaystyle \displaystyle x^{-\frac{2}{3}} \ = \ \frac{64}{9}\)
\(\displaystyle \displaystyle x^{\frac{2}{3}} \ = \ \left (\frac{3}{8}\right )^2\)
\(\displaystyle \displaystyle \left [ x^{\frac{2}{3}}\right ]^{\frac{3}{2}} \ = \ \left [\left (\frac{3}{8}\right )^2 \right ]^{\frac{3}{2}}\)
\(\displaystyle \displaystyle x \ = \ \left (\frac{3}{8}\right )^3 \)
What about the following?
\(\displaystyle x^{2/3} \ = \ \bigg(\dfrac{3}{8}\bigg)^2\)
\(\displaystyle (x^{2/3})^3 \ = \ \bigg[\bigg(\dfrac{3}{8}\bigg)^2\bigg]^3\)
\(\displaystyle x^2 \ = \ \bigg(\dfrac{3}{8}\bigg)^6\)
\(\displaystyle x \ = \ \pm\sqrt{\bigg(\dfrac{3}{8}\bigg)^6 \ }\)
\(\displaystyle x \ = \ \pm\bigg(\dfrac{3}{8}\bigg)^3 \)
\(\displaystyle x \ = \ \pm \dfrac{27}{512}\)
Cube root discussed in Wikipedia:
http://en.wikipedia.org/wiki/Cube_root
Formal definition
"The cube roots of a number x are the numbers y which satisfy the equation"
"If x and y are real, then there is a unique solution and so the cube root of a real number is sometimes defined by this equation.
If this definition is used, the cube root of a negative number is a negative number."
\(\displaystyle x^{2/3} \ = \ \bigg(\dfrac{3}{8}\bigg)^2\)
\(\displaystyle (x^{2/3})^3 \ = \ \bigg[\bigg(\dfrac{3}{8}\bigg)^2\bigg]^3\)
\(\displaystyle x^2 \ = \ \bigg(\dfrac{3}{8}\bigg)^6\)
\(\displaystyle x \ = \ \pm\sqrt{\bigg(\dfrac{3}{8}\bigg)^6 \ }\)
\(\displaystyle x \ = \ \pm\bigg(\dfrac{3}{8}\bigg)^3 \)
\(\displaystyle x \ = \ \pm \dfrac{27}{512}\)
Cube root discussed in Wikipedia:
http://en.wikipedia.org/wiki/Cube_root
Formal definition
"The cube roots of a number x are the numbers y which satisfy the equation"
"If x and y are real, then there is a unique solution and so the cube root of a real number is sometimes defined by this equation.
If this definition is used, the cube root of a negative number is a negative number."
Last edited:
\(\displaystyle \dfrac{1}{\sqrt[3]{x^2} \ } \ = \ \dfrac{64}{9} \ \ \ \ \ \ \ \ \ \ \ \ \)If this equation to the left is seen as an equivalent to the original equation, then it must be accepted that \(\displaystyle x \ = \ -\bigg(\dfrac{3}{8}\bigg)^3 \ \) is also a solution, as it checks in this equation.