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Number Systems

Short Answer Type

151. Simplify

fourth root of straight x to the power of bevelled 2 over 3 end exponent end root

and express the result in the exponential form of x.

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152. Find the value of ‘p’ if 5^p-3 × 3^{2p - 8} = 225.

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153. Simplify: $open parentheses 81 over 16 close parentheses to the power of negative begin inline style 3 over 4 end style end exponent cross times open parentheses 25 over 9 close parentheses to the power of negative 3 over 2 end exponent$

open parentheses 81 over 16 close parentheses to the power of negative begin inline style 3 over 4 end style end exponent cross times open parentheses 25 over 9 close parentheses to the power of negative 3 over 2 end exponent

equals open curly brackets open parentheses 3 over 2 close parentheses to the power of 4 close curly brackets to the power of negative 3 over 4 end exponent cross times open curly brackets open parentheses 5 over 2 close parentheses squared close curly brackets to the power of negative 3 over 2 end exponent equals space open parentheses 3 over 2 close parentheses to the power of 4 straight x open parentheses negative 3 over 4 close parentheses end exponent cross times open parentheses 5 over 3 close parentheses to the power of 2 straight x open parentheses negative 1 third close parentheses end exponent equals space open parentheses 3 over 2 close parentheses to the power of negative 3 end exponent cross times open parentheses 5 over 3 close parentheses to the power of 2 straight x open parentheses negative 3 over 2 close parentheses end exponent equals space open parentheses 3 over 2 close parentheses to the power of negative 3 end exponent cross times open parentheses 5 over 3 close parentheses to the power of negative 3 end exponent equals space open parentheses 3 over 2 cross times 5 over 3 close parentheses to the power of negative 3 end exponent equals space open parentheses 5 over 2 close parentheses to the power of negative 3 end exponent equals open parentheses 2 over 5 close parentheses cubed equals space 8 over 125

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154.

If a = 2, b = 3, then find the values of the following:

(i) (a^b + b^a)^-1 (ii) (a^a + b^b)^-1

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155.

Prove that $fraction numerator 2 to the power of 30 plus 2 to the power of 29 plus 2 to the power of 28 over denominator 2 to the power of 31 plus 2 to the power of 30 minus 2 to the power of 29 end fraction equals 7 over 10$

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156. Show that

x to the power of a left parenthesis b minus c right parenthesis end exponent over x to the power of b left parenthesis a minus c right parenthesis end exponent plus open parentheses x to the power of b over x to the power of a close parentheses to the power of c equals 1

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157. 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

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158. 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

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159.

Show that

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

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160. 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.

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