Respuesta :
Answer:
Option D
Step-by-step explanation:
Option A:
(3)¹ × (3)⁻¹⁰ = (3)⁽¹⁻¹⁰⁾ = 3⁹
= [tex]4\frac{1}{3^9}[/tex]
= [tex]4\frac{1}{19683}[/tex]
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
= [tex]4\frac{1}{3^3}[/tex]
= [tex]4\frac{1}{27}[/tex]
Therefore, option D is the answer.
The expression equivalent to [tex]\frac{1}{{27}}[/tex]is[tex]\boxed{{\mathbf{Option D}}-{{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{-7}}}}}[/tex].
Further explanation:
The expression is given as[tex]\frac{1}{{27}}[/tex].
Now, the above expression is simplified as follows:
[tex]\begin{aligned}\frac{1}{{27}}&=\frac{1}{{3 \times 3 \times 3}}\\&=\frac{1}{{{3^3}}}\\\end{aligned}[/tex]
Now, the power rule for rational exponent is given below.
The expression [tex]{a^{ - n}}[/tex] equivalent to expression [tex]\frac{1}{{{a^n}}}[/tex] that is[tex]\boxed{\frac{{\mathbf{1}}}{{\mathbf{a}}}{\mathbf{ = }}{{\mathbf{a}}^{{\mathbf{ - n}}}}}[/tex].
In the expression[tex]\frac{1}{{{3^3}}}[/tex], the value of [tex]a = 3[/tex]and[tex]n = 3[/tex].
The simplified form of the expression [tex]\frac{1}{{{3^3}}}[/tex] as follows:
[tex]\begin{aligned}\frac{1}{{{3^3}}}&={3^{-3}}\\&={3^{\left({-7+4}\right)}}\\\end{aligned}[/tex]
From the exponent rule[tex]\boxed{{{\mathbf{a}}^{{\mathbf{m + n}}}}{\mathbf{ = }}{{\mathbf{a}}^{\mathbf{m}}}{\mathbf{ \times }}{{\mathbf{a}}^{\mathbf{n}}}}[/tex], the above expression is evaluated as follows:
[tex]\begin{aligned}\frac{1}{{27}}&={3^{\left({-7+4}\right)}}\\&={3^{\left({ - 7} \right)}}\cdot {3^4}\\\end{aligned}[/tex]
Therefore, the expression equivalent to [tex]\frac{1}{{27}}[/tex]is[tex]\boxed{{{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{ - 7}}}}}[/tex].
Now, the four options are given below.
[tex]\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}[/tex]
Here, OPTION A is[tex]{{\text{3}}^1} \times {3^{ - 10}}[/tex].
The simplified form of OPTION A [tex]{{\text{3}}^1} \times {3^{ - 10}}[/tex] is given below.
[tex]\begin{aligned}{{\text{3}}^1}\times{3^{-10}}&={3^{1+\left({-10}\right)}}\\&={3^{1-10}}\\&={3^{-9}}\\\end{aliged}[/tex]
The above solution [tex]{3^{-9}}[/tex] of the expression [tex]{{\text{3}}^1}\times{3^{-10}}[/tex]does not matches with the obtained solution[tex]{3^4}\times{3^{-7}}[/tex].
The simplified form of OPTION B [tex]{{\text{3}}^{-1}}\times{3^{10}}[/tex] is given below.
[tex]\begin{aligned}{{\text{3}}^{-1}}\times{3^{10}}&={3^{-1+10}}\\&={3^9}\\\end{aligned}[/tex]
The above solution [tex]{3^9}[/tex] of the expression [tex]{{\text{3}}^{-1}}\times{3^{10}}[/tex]does not matches with the obtained solution[tex]{3^4}\times{3^{-7}}[/tex].
The simplified form of OPTION C [tex]{{\text{3}}^{-4}}\times{3^7}[/tex] is given below.
[tex]\begin{aligned}{{\text{3}}^{-4}}\times{3^7}&={3^{-4+7}}\\&={3^3}\\\end{aligned}[/tex]
The above solution [tex]{3^3}[/tex] of the expression [tex]{{\text{3}}^{-4}}\times{3^7}[/tex]does not matches with the obtained solution[tex]{3^4}\times{3^{-7}}[/tex].
OPTION D is given as [tex]{3^4}\times{3^{-7}}[/tex]and it matches with the obtained solution[tex]{3^4}\times{3^{-7}}[/tex].
From above four options, the expression equivalent to [tex]\frac{1}{{27}}[/tex]is[tex]{{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{ - 7}}}}[/tex].
Thus, the expression equivalent to [tex]\frac{1}{{27}}[/tex]is[tex]\boxed{{\mathbf{Option D}} - {{\mathbf{3}}^{\mathbf{4}}}{\mathbf{ \times }}{{\mathbf{3}}^{{\mathbf{ - 7}}}}}[/tex].
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Answer Details:
Grade: Junior High School
Subject: Mathematics
Chapter: Exponents and expressions
Keywords:expression, exponent, power exponent, equivalent, match, options,[tex]{3^4} \times {3^{ - 7}}[/tex],[tex]\frac{1}{{27}}[/tex]
