Tag: maths

Questions Related to maths

Which of the following represents the power of product rule?

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

  1. $(x\times y)^{a} = x^{a} \times y$

  2. $(x\times y)^{a} = x \times y^{a}$

  3. $(x\times y)^{a} = x^{a} + y^{a}$

  4. $(x\times y)^{a} = x^{a} \times y^{a}$

Show answer
Correct Option: D
Explanation:

If the product of the bases is powered  by the same exponent, 

then the result is multiplication of all the bases, 
each powered by the given exponent.
$\therefore (x\times y)^{a} = x^{a} \times y^{a}$.
So, option $D$ is correct.

Evaluate: $\left (\dfrac {2^{3}}{3^{3}} \right )^{2}$

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

  1. $\dfrac {64}{729}$

  2. $\dfrac {729}{64}$

  3. $\dfrac {32}{243}$

  4. $\dfrac {243}{32}$

Show answer
Correct Option: A
Explanation:

$\left (\dfrac {2^{3}}{3^{3}} \right )^{2} = \dfrac {2^{6}}{3^{6}} = \dfrac {64}{729}$


So, option $A$ is correct.

On simplifying  $\displaystyle 3^{3}\times a^{3}\times b^{3}$, we get

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

  1. $\displaystyle \left ( 3ab \right )^{3} $

  2. $\displaystyle 3\left ( ab \right )^{3} $

  3. $\displaystyle \left ( 27ab \right )^{3} $

  4. None of these

Show answer
Correct Option: A
Explanation:

$\displaystyle 3^{2}\times a^{3}\times b^{3}=\left ( 3ab \right )^{3}$.

This is the power of product law of exponents.
So, option $A$ is correct.

Find the value of: $\displaystyle \left [ \left ( -1 \right )^{2}\times \left ( -1 \right )^{3}\times \left ( -1 \right )^{4} \right ]^{6}$

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

  1. $3$

  2. $-1$

  3. $1$

  4. $\displaystyle 3^{6}$

Show answer
Correct Option: C
Explanation:

$\displaystyle \left [ \left ( -1 \right )^{2}\times \left ( -1 \right )^{3}\times \left ( -1 \right )^{4} \right ]^{6}$

=$\displaystyle \left [ \left ( -1 \right )^{12}\times \left ( -1 \right )^{18}\times \left ( -1 \right )^{24} \right ]$
=$\displaystyle \left [ 1\times 1\times 1 \right ]$ =$1$
So, option $C$ is correct.

The value of $\displaystyle \left (-4  \right ) ^{3}\times \left ( -3 \right )^{3}$ is _____?

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

  1. $\displaystyle 12^{3}$

  2. $\displaystyle -12^{3}$

  3. $\displaystyle -7^{3}$

  4. $\displaystyle 7^{3}$

Show answer
Correct Option: A
Explanation:

$\displaystyle \left ( -4 \right )^{3}\times \left ( -3 \right )^{3}=\left ( -4\times -3 \right )^{3}$

$=\displaystyle \left ( -4\times -3 \right )^{3}$
$=\displaystyle 12^3$

Find the expression which equals $\displaystyle a^{x}\times b^{x}$.

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

  1. $\left [\displaystyle a^{x}+ b^{x} \right ]$

  2. $\displaystyle \left ( ab\right )^x $

  3. $\displaystyle \left (a+b \right )^{x} $

  4. $\displaystyle a\left ( b \right )^{x} $

Show answer
Correct Option: B
Explanation:

$\displaystyle a^{x}\times b^{x}=\left ( ab \right )^{n}$

So, option $B$ is correct.

Evaluate: $\displaystyle 5^{2}\times 3^{2} $

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

  1. $\displaystyle \left (53 \right ) ^{2}$

  2. $\displaystyle \left (15 \right ) ^{2}$

  3. $\displaystyle \left (8 \right ) ^{2}$

  4. $60$

Show answer
Correct Option: B
Explanation:

$\displaystyle 5^{2}\times 3^{2}=\left ( 15 \right )^{2}$

So, option $B$ is correct.

Evaluate: $\displaystyle \left [ \left ( 4 \right )^{\tfrac{1}{4}}\times \left ( 2 \right )^{\tfrac{1}{2}}\times \left ( 5 \right )^{\tfrac{1}{5}} \right ]^{0}$

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

  1. $40$

  2. $0$

  3. $1$

  4. $10$

Show answer
Correct Option: C
Explanation:

=$\displaystyle \left [ \left ( 4 \right )^{\tfrac{1}{4}}\times \left ( 2 \right )^{\tfrac{1}{2}} \times \left ( 5 \right )^{\tfrac{1}{2}}\right ]^{0}$

=$\displaystyle 4^{0}\times 2^{0}\times 5^{0}$
=$\displaystyle 1\times 1\times 1$
=$1$
So, option $C$ is correct.

The value of $\displaystyle \left [ \left ( \frac{-2}{5} \right )^{3} \right ]^{2}$ is:

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

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

  2. $\displaystyle -\frac{32}{3125}$

  3. $\displaystyle\frac{64}{15625}$

  4. $\displaystyle -\frac{64}{15625}$

Show answer
Correct Option: C
Explanation:

$\displaystyle \left [ \left ( \frac{-2}{5} \right )^{3} \right ]^{2}=\left ( \frac{-2}{5} \right )^{6}=\frac{\left ( -2 \right )^{6}}{5^{6}}=\frac{64}{15625}$

Hence, option $C$ is correct.

Simplify: $\displaystyle \left ( -a \right )^{9}\times \left ( -b \right )^{9}$ 

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

  1. $\displaystyle \left ( ab \right )^{9}$

  2. $\displaystyle \left ( -ab \right )^{9}$

  3. $\displaystyle -\left ( ab \right )^{9}$

  4. $\displaystyle \left ( a-b \right )^{9}$

Show answer
Correct Option: A
Explanation:

$(-a)^9 \times (-b)^9 = [ -a \times -b]^9$


= $ [ a \times b]^9$

=$ (ab)^9$
So, option $A$ is correct.