Question

Which expressions are equivalent to \dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5} 5 1 ​ ⋅ 5 1 ​ ⋅ 5 1 ​ ⋅ 5 1 ​ start fraction, 1, divided by, 5, end fraction, dot, start fraction, 1, divided by, 5, end fraction, dot, start fraction, 1, divided by, 5, end fraction, dot, start fraction, 1, divided by, 5, end fraction ? Choose 2 answers: Choose 2 answers: (Choice A) A (5^{-2})^{2}(5 −2 ) 2 left parenthesis, 5, start superscript, minus, 2, end superscript, right parenthesis, squared (Choice B) B (5^{-4})^{0}(5 −4 ) 0 left parenthesis, 5, start superscript, minus, 4, end superscript, right parenthesis, start superscript, 0, end superscript (Choice C) C \dfrac{5^1}{5^4} 5 4 5 1 ​ start fraction, 5, start superscript, 1, end superscript, divided by, 5, start superscript, 4, end superscript, end fraction (Choice D) D 5^2\cdot 5^{-6}5 2 ⋅5 −6

Answer 1

Given:

The expression is

$\dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}$

To find:

The expressions which are equivalent to the given expression.

Solution:

We have,

$\dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}=\dfrac{1}{5^4}$

$\dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}\cdot \dfrac{1}{5}=5^{-4}$

In option A,

$(5^{-2})^2=5^{-2\times 2}$

$(5^{-2})^2=5^{-4}$

This expression is equivalent to the given expression.

In option B,

$(5^{-4})^0=5^{-4\times 0}$

$(5^{-4})^0=5^{0}\neq 5^{-4}$

This expression is not equivalent to the given expression.

Option C,

$\dfrac{5^1}{5^4}=5^{1-4}$

$\dfrac{5^1}{5^4}=5^{-3}\neq 5^{-4}$

This expression is not equivalent to the given expression.

Option D,

$5^2\cdot 5^{-6}=5^{2-6}$

$5^2\cdot 5^{-6}=5^{-4}$

This expression is equivalent to the given expression.

Therefore, the correct options are A and D.