Tag: powers and exponents

Questions Related to powers and exponents

$For \ a \  natural \  number \  n , \  2n(n-1)!\leqslant  n^n, if$

maths powers and exponents scientific notation use of exponents power of 10

  1. $n<2$

  2. $n>2$

  3. $n\ge2$

  4. Never

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Correct Option: C

Which of the following represents the given expression?
$a^2b^3\times 2ab^2$ ?

maths powers and exponents scientific notation use of exponents power of 10

  1. $2a^3b^4$

  2. $2a^3b^5$

  3. $2ab$

  4. $a^3b^5$

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Correct Option: B
Explanation:
We know that if a number say $b$ is multiplied three times. That is, $b\times b\times b$ can be written as $b^3$.
=> $b^{3}$ = $b\times b\times b$
Similarly,
$a^{2}b^3$ = $a \times a\times b\times b\times b$
$2ab^{2}$ = $ 2 \times a\times b\times b$.
Thus, 
$a^{2}b^{3}\times 2ab^{2} = a \times a\times b\times b\times b \times 2 \times a\times b\times b $.
                      $= 2 \times a \times a \times a\times b\times b\times b\times b\times b$.
                      $ = 2a^{3}b^{5}$.

If $\displaystyle 2^{2^{3}}=j, 2^{3^{2}}=k, 3^{2^{2}}=\varphi ,$ then 

maths powers and exponents scientific notation use of exponents power of 10

  1. $k = 2j$

  2. $j < k$

  3. $\displaystyle \varphi < k$

  4. All of these

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Correct Option: D
Explanation:

  $j=2^{2{^3}}$, $k=2^{3^{2}}$, $\varphi=3^{2^{2}}$

$\Rightarrow$  $j=256,\,\,k=512,\,\,\varphi=81$
$\Rightarrow$  Option A says $k=2j$, which is true because $j=256$, so $k=2\times 256=512$
$\Rightarrow$  Option B says $j<k$, which is true because value of $j$ is $255$ and value of $k$ is $512$
$\Rightarrow$   Option C says $\varphi <k$, which is true because value of $\varphi$ is $81$ and value of $k$ is $512$

If $\displaystyle a^{m}=b^{m}$ and $(m > 0)$, then which of the following options could be true:

maths powers and exponents scientific notation use of exponents power of 10

  1. $a = -b$

  2. $a  + b= 0$

  3. $2a - b = 0$

  4. $\displaystyle \dfrac{a^{2}}{b^{2}}=1$

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Correct Option: D
Explanation:

Given $a^m=b^m$


Dividing by $b^m$

$\dfrac{a^m}{b^m}=1$
${\left(\dfrac{a}{b}\right)}^m=1$

Only Option D satisfies this answer.

Use an appropriate comparison symbol $0.00000998$ ______ $0.0000116$.

maths powers and exponents scientific notation use of exponents power of 10

  1. $<$

  2. $>$

  3. $=$

  4. None of these

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Correct Option: A
Explanation:

$\displaystyle 0\cdot 00000998= 9\cdot 89\times 10^{-6}$
$\displaystyle 0\cdot 0000116= 1\cdot 16\times 10^{-5}$
Thus $\displaystyle 9\cdot 89\times 10^{-6}< 1\cdot 16\times 10^{-5}$

$0.000008$ _______ $0.000016$

maths powers and exponents scientific notation use of exponents power of 10

  1. is half of

  2. is double than

  3. is one-fourth of

  4. is one-third of

Show answer
Correct Option: A
Explanation:

$\displaystyle 0\cdot 000008= 8\times 10^{-6}$


$\displaystyle 0\cdot 000016= 1\cdot 6\times 10^{-5}$
Now
$\displaystyle \frac{8\times 10^{-6}}{1\cdot 6\times 10^{-5}}= \frac{8}{1\cdot 6}\times 10^{-1}= 5\times 10^{-1}=0\cdot 5$
$\displaystyle \therefore 8\times 10^{-6}$ is half of $\displaystyle 1\cdot 6\times 10^{-5}$

The thickness of paper is $0.004$ m and that of another paper is $0.008$ m. Compare their sizes.

maths powers and exponents scientific notation use of exponents power of 10

  1. Paper-1 is double thicker than paper-2.

  2. Thickness of paper-1 is half than paper-2.

  3. Thickness of paper-1 is one-fourth than paper-2.

  4. Paper-1 and paper-2 are both equally thick.

Show answer
Correct Option: B
Explanation:

$\displaystyle 0\cdot 004= 4\times 10^{-3}$ m


$\displaystyle 0\cdot 008=8\times 10^{-3}$
Now
$\displaystyle \frac{0\cdot 004}{0\cdot 008}= \frac{4\times 10^{-3}}{8\times 10^{-3}}= \frac{4}{8}= \frac{1}{2}$
$\displaystyle \therefore $ Thickness of paper-1 is half than paper-2

Compare the folllowing:

$0.000000038$ _______ $\displaystyle 3\cdot 8\times 10^{-8}$

maths powers and exponents scientific notation use of exponents power of 10

  1. $<$

  2. $>$

  3. $=$

  4. None of these

Show answer
Correct Option: C
Explanation:

$\displaystyle 0\cdot 000000038= 3\cdot 8\times 10^{-8}$

Use an appropriate comparison symbol $0.0000486$ _____ $0.00000387$.

maths powers and exponents scientific notation use of exponents power of 10

  1. $<$

  2. $>$

  3. $=$

  4. None of these

Show answer
Correct Option: B
Explanation:

$\displaystyle 0.0000486=4\cdot 86\times 10^{-5}$
$\displaystyle 0.00000387=3\cdot 87\times 10^{-6}$
Thus
$\displaystyle 4\cdot 86\times 10^{-5}> 3\cdot 87\times 10^{-6}$

If $2^{p + 2} + 2^{p + 1} = 96$, then find the value of $ p$.

maths powers and exponents scientific notation use of exponents power of 10

  1. $1$

  2. $2$

  3. $3$

  4. $4$

  5. $5$

Show answer
Correct Option: D
Explanation:

As given, $2^{p+2}+2^{p+1}=96$
Rewriting the left hand side of equation using $x^a.x^b=x^{a+b}$.
$2^p.2^2+2^p.2^1=96$
$2^p(2^2+2^1)=96$
$2^p\times6=96$
$2^p=16=2^4$
$p=4$
Hence, option D is correct.