Verified Answer :

Answer:

Option D

Step-by-step explanation:

Option A:

(3)¹ × (3)⁻¹⁰   =  (3)⁽¹⁻¹⁰⁾   = 3⁹

= \(4\frac{1}{3^9}\)

= \(4\frac{1}{19683}\)

Option B :

(3)⁻¹ × (3) ¹⁰   = 3⁽¹⁰⁻¹⁾

= 3⁹

= 19683

Option C :

(3⁻⁴) × (3)⁷   = 3⁽⁷⁻⁴⁾

               = 3³

Option D :

3⁴ × 3⁻⁷ = 3⁴⁻⁷

= 3⁻³

=  \(4\frac{1}{3^3}\)

= \(4\frac{1}{27}\)

Therefore, option D is the answer.

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\(\boxed{{\mathbf{Option D}} - {{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{ - 7}}}}}\).

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Answer Details:

Grade: Junior High School

Subject: Mathematics

Chapter: Exponents and expressions

Keywords:expression, exponent, power exponent, equivalent, match, options,\({3^4} \times {3^{ - 7}}\),\(\frac{1}{{27}}\)

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