Tag: powers and exponents

Questions Related to powers and exponents

If $8^x = 16^{x-1}$, find $x $.

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

  1. $\dfrac{1}{8}$

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

  3. $2$

  4. $4$

Show answer
Correct Option: D
Explanation:

Given, $8^x=(16)^{x-1}$

Left hand side, $8^x=(2^3)^x=2^{3x}$
Right hand side $16^{x-1}=(2^4)^{x-1}=2^{4(x-1)}=2^{(4x-4)}$
Equating both sides, we get
$2^{3x}=2^{{4x-4}}$
As bases are equal, then powers must be equal, so
$\Rightarrow 3x=4x-4$
$ \Rightarrow x=4$

If $a^{b} = 4  -ab$ and $b^{a} = 1$, where $a$ and $b$ are positive integers, find $a$.

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

  1. $0$

  2. $1$

  3. $2$

  4. $3$

Show answer
Correct Option: C
Explanation:
Plug in real numbers for $a$ and $b$. Since it isn’t clear what numbers to plug in to satisfy the first equation, look at the second equation instead. First, realize that  $a$  cannot be  $0$  since a is a positive integer.
Since, $a\neq0$, so the only way to get $b^a=1$ is if $b=1 $(As $1$ to any power is $1$).
Plugging $b=1$ in to the first equation,
$a^b=4-ab$
$a^1=4-a\times1$
$a=4-a$
$a=2$
Hence, option C is correct.

If $9^n = 27^{n+1}$, then calculate the value of $2^n $.

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

  1. $-\dfrac{10}{3}$

  2. $-\dfrac{8}{3}$

  3. $-\dfrac{3}{8}$

  4. $\dfrac{1}{8}$

  5. $\dfrac{3}{8}$

Show answer
Correct Option: D
Explanation:

Given ${ 9 }^{ n }={ 27 }^{ n+1 }$, which implies
$ { 3 }^{ 2n }={ 3 }^{ 3n+3 }$
Now compare powers, we get
$2n = 3n+3$ , which implies $n = -3$
Therefore ${ 2 }^{ n }={ 2 }^{ -3 }=\dfrac { 1 }{ 8 } $

If $4^{2x + 2} = 64$, then calculate the value of $x $.

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

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

  2. $1$

  3. $\dfrac {3}{2}$

  4. $2$

Show answer
Correct Option: A
Explanation:

Given is $4^{2x+2}=64$

LHS: 
$\Rightarrow 4^{ 2x+2 }=(2^{ 2 })^{ 2x+2 }\ \Rightarrow { 2 }^{ 4x+4 }$
RHS: 
$\Rightarrow 64=2^6$

Now, LHS $=$ RHS
$\Rightarrow { 2 }^{ 4x+4 }={ 2 }^{ 6 }$
As bases are equal, so powers must be equal,
$\Rightarrow 6=4x+4$
$\Rightarrow 4x=2\ \Rightarrow x=\dfrac { 1 }{ 2 } $

If $x$ is a positive integer satisfying $x^7=k$ and $x^9=m$, which of the following must be equal to $x^{11}$?

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

  1. $\cfrac{m^2}{k}$

  2. $m^2-k$

  3. $m^2-7$

  4. $2k-\cfrac{m}{3}$

  5. $k+4$

Show answer
Correct Option: A
Explanation:
Given, $x^7=k, x^9=m$
$x^{11}=\dfrac{x^{18}}{x^{7}}$ 
Put the given values, we get
$x^{11}=\dfrac{m^{2}}{k}$

If $n$ and $k$ are positive integers and $8^n=2^k$, what is the value of $\dfrac{n}{k}$?

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

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

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

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

  4. $3$

Show answer
Correct Option: B
Explanation:

Given 

${8}^{n}$ $=$ ${2}^{k}$
As $'8'$ can be written as $'$${2}^{3}$$'$,
${({2}^{3})}^{n}$ $=$ ${2}^{k}$
${2}^{3 \space \times \space n}$ $=$ ${2}^{k}$
${2}^{3n}$ $=$ ${2}^{k}$
Equating the exponents as the bases are same, we get
$3n$ $=$ $k$
Rearranging the terms, we get
$\dfrac {n}{k}$ $=$ $\dfrac {1}{3}$
Therefore, the value of $\dfrac {n}{k}=\dfrac {1}{3}$.

If $7^n\times 7^3 = 7^{12}$, what is the value of $n$?

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

  1. $2$

  2. $4$

  3. $9$

  4. $15$

  5. $36$

Show answer
Correct Option: C
Explanation:

Given, ${7}^{n}$ $\times$ ${7}^{3}$ $=$ ${7}^{12}$

Rearranging terms, we get
${7}^{n}$ $=$ $\dfrac {{7}^{12}}{{7}^{3}}$
We know that
$\dfrac {{x}^{a}}{{x}^{b}}$ $=$ ${x}^{a \space - \space b}$
Hence, ${7}^{n}$ $=$ ${7}^{12 \space - \space 3}$
$\therefore {7}^{n}$ $=$ ${7}^{9}$
Again rearranging terms, we get
$n$ $=$ $\log _{7}{{7}^{9}}$
We know that
$\log _{x}{{x}^{a}}$ $=$ $a$$\log _{x}{x}$ and $\log _{x}{x}$ $=$ $1$
Hence, $n$ $=$ $9$$\log _{7}{7}$
$\therefore n$ $=$ $9$ $\times$ $1$
$\therefore n$ $=$ $9$
Therefore, the value of $'n'$ is $'9'$.

Which of the following has the greatest value?

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

  1. $(6^{2} \times 6)^{4}$

  2. $(36)^{5}$

  3. $(36^{2} \times 6^{3})^{2}$

  4. $(216)^{4}$

  5. $(6^{4})^{4}$

Show answer
Correct Option: E
Explanation:

  1. ${ ({ 6 }^{ 2 }\times 6) }^{ 4 }={ { (6 }^{ 3 }) }^{ 4 }={ 6 }^{ 12 }$
  2. ${ 36 }^{ 5 }={ ({ 6 }^{ 2 }) }^{ 5 }={ 6 }^{ 10 }$
  3. ${ ({ 36 }^{ 2 }\times { 6 }^{ 3 }) }^{ 2 }={ ({ 6 }^{ 4 }\times { 6 }^{ 3 }) }^{ 2 }={ ({ 6 }^{ 7 }) }^{ 2 }={ 6 }^{ 14 }$
  4. ${ 216 }^{ 4 }={ ({ 6 }^{ 3 }) }^{ 4 }={ 6 }^{ 12 }$
  5. ${ ({ 6 }^{ 4 }) }^{ 4 }={ 6 }^{ 16 }$
  • Therefore option $E$ has maximum value

If $3^{n} = n^{6}$, find the value of $ n^{18} $

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

  1. $3^{n} n^{3}$

  2. $3^{n} n^{12}$

  3. $9^{n}$

  4. $3^{12n}$

Show answer
Correct Option: B
Explanation:
Given, $3^n=n^6$
We need to find the value of $n^{18}$
$\therefore {n}^{18}= n^6. n^{12}=3^n.n^{12}$
$\therefore n^{6+12}=3^n. n^{12}$
$\therefore n^{18}= {3}^{n}{n}^{12}$

If $64^{x} = 4^{x^{2} - 4}$, then find the value of $x$.

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

  1. $x = 4$ or $x = -1$

  2. $x = -4$ or $x = 1$

  3. $x = 10$

  4. $x = \sqrt {20}$

  5. $x = 3$

Show answer
Correct Option: A
Explanation:
  • ${ 64 }^{ x }={ 4 }^{ 3x }={ 4 }^{ { x }^{ 2 }-4 }$ , by equating powers , we get,
  • ${ x }^{ 2 }-4 = 3x$ , which implies ${ x }^{ 2 }-3x-4 = 0$
  • $\Rightarrow x^2-4x+x-4=0$
  • $\Rightarrow x(x-4)+1(x-4)= 0 $
  • $\Rightarrow (x-4)(x+1)=0$
  • The roots are $x=4,-1$