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Class 10 Class 12

Number Systems

Write the following in the ascend ing order of their magnitude

fourth root of 3 comma space cube root of 2 comma space cube root of 4.

fourth root of 3 equals 3 to the power of 1 fourth end exponent equals 3 to the power of 3 over 12 end exponent equals left parenthesis 3 cubed right parenthesis to the power of 1 over 12 end exponent equals left parenthesis 27 right parenthesis to the power of 1 over 12 end exponent cube root of 2 equals 2 to the power of 1 third end exponent equals 2 to the power of 4 over 12 end exponent equals left parenthesis 2 to the power of 4 right parenthesis to the power of 1 over 12 end exponent equals left parenthesis 16 right parenthesis to the power of 1 over 12 end exponent cube root of 4 equals 4 to the power of 1 third end exponent equals 4 to the power of 4 over 12 end exponent equals left parenthesis 4 to the power of 4 right parenthesis to the power of 1 over 12 end exponent equals left parenthesis 256 right parenthesis to the power of 1 over 12 end exponent because space space space 16 less than 27 less than 256 therefore space space left parenthesis 16 right parenthesis to the power of 1 over 12 end exponent less than left parenthesis 27 right parenthesis to the power of 1 over 12 end exponent less than left parenthesis 256 right parenthesis to the power of 1 over 12 end exponent rightwards double arrow space space space space cube root of 2 space less than space fourth root of 3 less than cube root of 4

Hence, the ascending order is

cube root of 2 comma space fourth root of 3 comma cube root of 4

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Simplify:

fraction numerator 9 to the power of begin display style 1 third end style end exponent cross times 27 to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent cross times 3 to the power of begin display style 1 third end style end exponent end fraction space space space

fraction numerator 9 to the power of begin display style 1 third end style end exponent cross times 27 to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent cross times 3 to the power of begin display style 1 third end style end exponent end fraction space space space space space equals space fraction numerator left parenthesis 3 squared right parenthesis to the power of begin display style 1 third end style end exponent cross times left parenthesis 3 cubed right parenthesis to the power of begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 plus 1 third end style end exponent end fraction space space space space space equals space fraction numerator 3 to the power of 2 cross times begin display style 1 third end style end exponent straight x 3 to the power of 3 straight x begin display style 1 half end style end exponent over denominator 3 to the power of begin display style 1 over 6 end style end exponent end fraction space space space space equals space 3 to the power of begin display style 2 over 3 plus 3 over 2 end style end exponent over 3 to the power of begin display style 1 over 6 end style end exponent space space space space equals space space 3 to the power of begin display style 13 over 6 end style end exponent over 3 to the power of begin display style 1 over 6 end style end exponent space space space space equals space 3 squared space space space space equals space 9

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Show that

(x^{a - b})^{a + b}. (x^{b - c})^{b + c}. (x^{c - a})^{c + a} = 1

(x^{a - b})^{a + b}. (x^{b - c})^{b + c}. (x^{c - a})^{c + a}= x^{(a - b) (a + b)}. x^{(b - c) (b + c)}. x^{(c - a) (c + a)}= x^{a² - b²}. x^{b² - c²}. x^{c² - a²}= x^{a² - b² b² + b² - c² + c² - a²}= X⁰= 1

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Simplify :
17².17⁵

Solution is not provided.

Ans. 17⁷

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Simplify:

open curly brackets 5 left parenthesis 8 to the power of 1 third end exponent plus 27 to the power of 1 third end exponent right parenthesis cubed close curly brackets to the power of 1 fourth end exponent space space space space space space

open curly brackets 5 left parenthesis 8 to the power of 1 third end exponent plus 27 to the power of 1 third end exponent right parenthesis cubed close curly brackets to the power of 1 fourth end exponent space space space space space space

equals open square brackets 5 open curly brackets left parenthesis 2 cubed right parenthesis to the power of 1 third end exponent plus left parenthesis 3 cubed right parenthesis close curly brackets close square brackets to the power of 1 fourth end exponent

equals left parenthesis 5 to the power of 1 5 cubed right parenthesis to the power of 1 fourth end exponent equals left parenthesis 5 to the power of 1 plus 3 end exponent right parenthesis to the power of 1 fourth end exponent equals left parenthesis 5 to the power of 4 right parenthesis to the power of 1 fourth end exponent equals 5 to the power of 4 straight x 1 fourth end exponent equals 5 to the power of 1 equals 5

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