Tag: maths

Questions Related to maths

If $\displaystyle \log _{16} 8$ = $\displaystyle \frac {3}{m}$, then value of $m$ is equal to 

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
Correct Option: D
Explanation:
$\log _{16}8=\dfrac{3}{m}$

$\therefore \dfrac{\log 8}{\log 16}=\dfrac{3}{m}$      ....($\log _ba=\dfrac{\log a}{\log b}$)

$\therefore \dfrac{\log 2^3}{\log 2^4}=\dfrac{3}{m}$   ...($\log a^b=a\log b$)

$\therefore \dfrac{3}{m}=\dfrac{3}{4}$ $ \Rightarrow m = 4$.

Match the numbers in column-I with the rules in column- II

No Column-I No Column-II
1 30 a $n^3+n/2 $
2 63 b $3n^2+3$
3 66 c $n^3+4$
4 110 d $n^2-2n$
5 127 e $n^3-3n$
f $2n^2-1$


Which rule the number 30 follows?

maths power and exponent power of powers laws of exponents and powers law of indices

  1. b

  2. c

  3. d

  4. e

Show answer
Correct Option: A
Explanation:

Substitute the value of $n=3$ in $b$
 $\Rightarrow  3n^2+3=3\times 3^2+3=27+3= 30 $
Hence, option 'A' is correct.

$\displaystyle (64)^{-\tfrac{1}{2}}-(-32)^{-\tfrac{4}{5}}=?$

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $\displaystyle \frac{1}{8}$

  2. $\displaystyle \frac{3}{8}$

  3. $\displaystyle \frac{1}{16}$

  4. $\displaystyle \frac{3}{16}$

Show answer
Correct Option: C
Explanation:

$\displaystyle (64) ^{-\tfrac{1}{2}}-(-32)^{-\tfrac{4}{5}}$
$=\left({2^6}\right)^{\frac{-1}{2}}-\left({(-2)^5}\right)^{\frac{-4}{5}}$
$=(2)^{ -\tfrac { 6 }{ 2 }  }-(-2)^{ -\tfrac { 4\times 5 }{ 5 }  }$
$=(2)^{ -3 }-(-2)^{ -4 }$
$\dfrac { 1 }{ 8 } -\dfrac { 1 }{ 16 } =\dfrac { 1 }{ 16 } $
Answer $C$ option, $ \cfrac { 1 }{ 16 } $

If $a^x=\sqrt{b},b^y = \sqrt [3]{c}$ and $c^z = \sqrt {a}$ then the value of $xyz$

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $\displaystyle \frac {1}{2}$

  2. $\displaystyle \frac {1}{3}$

  3. $\displaystyle \frac {1}{6}$

  4. $\displaystyle \frac {1}{12}$

Show answer
Correct Option: D
Explanation:

$a^x=\sqrt{b}=b^{1/2}$
$a=b^{1/2x}$
$b^y=\sqrt[3]{c}$
$b^y=c^{1/3}$
$b=c^{1/3y}$
$c^z=a^{1/2}$
$c=a^{1/2z}=b^{1/4xz}$
$c^1=c^{1/12xyz}$
$\displaystyle 1= \frac {1}{12xyz}$
$\displaystyle xyz = \frac {1}{12}$

Solve for x ; $\displaystyle \frac{2^{x-3}}{8^{-x}} = \frac{32}{4^{(1/2)x}}$

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $2\displaystyle \frac{1}{5}$

  2. $1\displaystyle \frac{1}{5}$

  3. $3\displaystyle \frac{1}{5}$

  4. $1\displaystyle \frac{3}{5}$

Show answer
Correct Option: D
Explanation:

 $\displaystyle \frac{2^{x-3}}{8^{-x}} = \frac{32}{4^{(1/2)x}}$
$\frac { 2^{ x-3 } }{ 2^{ -3x } } =\frac { { 2 }^{ 5 } }{ 2^{ (2/2)x } } $\ 
${ 2 }^{ x-3+3x }={ 2 }^{ 5-x }$
$4x-3=5-x$
$5x=8$
$x=1\frac { 3 }{ 5 } $
Answer (D)  $1\frac { 3 }{ 5 } $

If $2^a\,>\,4^c\;and\;3^b\,>\,9^a\;and\;a,\,b,\,c$ all positive, then

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $c\,<\,a\,<\,b$

  2. $b\,<\,c\,<\,a$

  3. $c\,<\,b\,<\,a$

  4. $a\,<\,b\,<\,c$

Show answer
Correct Option: A
Explanation:

$2^a\,>\,4^c\;\;\;\;\;\;3^b\,>\,9^a$
$2^a\,>\,2^{2c}\;\;\;\;\;3^b\,>\,3^{2a}$
$a\,>\,2c\;\;\;\;\;\;\;b\,>\,2a$
$\therefore\;a\,>\,c-(i)\;\;\therefore\;b\,>\,a-(ii)$
From (1) & (2), we have
$c\,<\,a\,<\,b$

Find the value of: $[(-2)^{3} \times (-2)^{-4}]^{2}$

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $4$

  2. $\dfrac {1}{4}$

  3. $-4$

  4. $-\dfrac {1}{4}$

Show answer
Correct Option: B
Explanation:

$[(-2)^{3} \times (-2)^{-4}]^{2} = [(-2)^{3-4}]^{2}$

$=[(-2)^{-1}]^{2}$
$=\left (\dfrac {1}{-2}\right )^{2}$$= \dfrac {1}{4}$

So, option $B$ is correct.

Find m so that $\displaystyle \left ( \frac{11^{2}}{13^{2}} \right )^{-6}=\left ( \frac{13}{11} \right )^{m}$

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $-12$

  2. $-6$

  3. $6$

  4. $12$

Show answer
Correct Option: D
Explanation:

$\displaystyle \left ( \frac{11^{2}}{13^{2}} \right )^{-6}=\left ( \frac{11}{13} \right )^{-12}=\left ( \frac{13}{11} \right )^{12}$
Also,$\displaystyle \left ( \frac{13}{11} \right )^{12}=\left ( \frac{13}{11} \right )^{m}$
$\displaystyle \therefore m=12$

Find the value of: $[(-3)^{-4} \div (-3)^{-5}]^{3}$

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $-27$

  2. $27$

  3. $\dfrac {1}{27}$

  4. $-\dfrac {1}{27}$

Show answer
Correct Option: A
Explanation:

$[(-3)^{-4} \div (-3)^{-5}]^{3} = [(-3)^{-4+5}]^{3}$

$=[(-3)^{1}]^{3}$ $= (-3)^{3}$
$= -27$

So, option $A$ is correct.

The value of $(6^{4} \times 7^{2})^{\tfrac {1}{2}}$ is equal to _____

maths power and exponent power of powers laws of exponents and powers law of indices

  1. $49$

  2. $42$

  3. $252$

  4. $36$

Show answer
Correct Option: C
Explanation:

$(6^{4} \times 7^{2})^{\tfrac {1}{2}} = 6^{\tfrac {4}{2}} \times 7^{\tfrac {2}{2}}$
$= 6^{2} \times 7$
$= 36\times 7$
$= 252$

So, option $C$ is correct.