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4 years ago

Which expression is equivalent to 1/27 ( 1 over 27 ) ?

2 answers:

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The expression equivalent to $\frac{1}{{27}}$ is $\boxed{{\mathbf{Option D}}-{{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{-7}}}}}$ .

Further explanation:

The expression is given as $\frac{1}{{27}}$ .

Now, the above expression is simplified as follows:

$\begin{aligned}\frac{1}{{27}}&=\frac{1}{{3 \times 3 \times 3}}\\&=\frac{1}{{{3^3}}}\\\end{aligned}$

Now, the power rule for rational exponent is given below.

The expression ${a^{ - n}}$ equivalent to expression $\frac{1}{{{a^n}}}$ that is $\boxed{\frac{{\mathbf{1}}}{{\mathbf{a}}}{\mathbf{ = }}{{\mathbf{a}}^{{\mathbf{ - n}}}}}$ .

In the expression $\frac{1}{{{3^3}}}$ , the value of $a = 3$ and $n = 3$ .

The simplified form of the expression $\frac{1}{{{3^3}}}$ as follows:

$\begin{aligned}\frac{1}{{{3^3}}}&={3^{-3}}\\&={3^{\left({-7+4}\right)}}\\\end{aligned}$

From the exponent rule $\boxed{{{\mathbf{a}}^{{\mathbf{m + n}}}}{\mathbf{ = }}{{\mathbf{a}}^{\mathbf{m}}}{\mathbf{ \times }}{{\mathbf{a}}^{\mathbf{n}}}}$ , the above expression is evaluated as follows:

$\begin{aligned}\frac{1}{{27}}&={3^{\left({-7+4}\right)}}\\&={3^{\left({ - 7} \right)}}\cdot {3^4}\\\end{aligned}$

Therefore, the expression equivalent to $\frac{1}{{27}}$ is $\boxed{{{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{ - 7}}}}}$ .

Now, the four options are given below.

$\begin{aligned}&{\text{OPTION A}}\to{{\text{3}}^1}\times {3^{ - 10}}\hfill\\&{\text{OPTION B}}\to{{\text{3}}^{-1}}\times {3^{10}}\hfill\\&{\text{OPTION C}}\to{{\text{3}}^{-4}}\times {3^7}\hfill\\&{\text{OPTION D}}\to{{\text{3}}^4}\times{3^{-7}}\hfill\\\end{aligned}$

Here, OPTION A is ${{\text{3}}^1} \times {3^{ - 10}}$ .

The simplified form of OPTION A ${{\text{3}}^1} \times {3^{ - 10}}$ is given below.

$\begin{aligned}{{\text{3}}^1}\times{3^{-10}}&={3^{1+\left({-10}\right)}}\\&={3^{1-10}}\\&={3^{-9}}\\\end{aliged}$

The above solution ${3^{-9}}$ of the expression ${{\text{3}}^1}\times{3^{-10}}$ does not matches with the obtained solution ${3^4}\times{3^{-7}}$ .

The simplified form of OPTION B ${{\text{3}}^{-1}}\times{3^{10}}$ is given below.

$\begin{aligned}{{\text{3}}^{-1}}\times{3^{10}}&={3^{-1+10}}\\&={3^9}\\\end{aligned}$

The above solution ${3^9}$ of the expression ${{\text{3}}^{-1}}\times{3^{10}}$ does not matches with the obtained solution ${3^4}\times{3^{-7}}$ .

The simplified form of OPTION C ${{\text{3}}^{-4}}\times{3^7}$ is given below.

$\begin{aligned}{{\text{3}}^{-4}}\times{3^7}&={3^{-4+7}}\\&={3^3}\\\end{aligned}$

The above solution ${3^3}$ of the expression ${{\text{3}}^{-4}}\times{3^7}$ does not matches with the obtained solution ${3^4}\times{3^{-7}}$ .

OPTION D is given as ${3^4}\times{3^{-7}}$ and it matches with the obtained solution ${3^4}\times{3^{-7}}$ .

From above four options, the expression equivalent to $\frac{1}{{27}}$ is ${{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{ - 7}}}}$ .

Thus, the expression equivalent to $\frac{1}{{27}}$ is $\boxed{{\mathbf{Option D}} - {{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{ - 7}}}}}$ .