Respuesta :
we know that
A polynomial in the form [tex]a^{3}-b^{3}[/tex] is called adifference of cubes. Both terms must be a perfect cubes
Let's verify each case to determine the solution to the problem
case A) [tex]9w^{33} -y^{12}[/tex]
we know that
[tex]9=3^{2}[/tex] ------> the term is not a perfect cube
[tex]w^{33}=(w^{11})^{3}[/tex] ------> the term is a perfect cube
[tex]y^{12}=(y^{4})^{3}[/tex] ------> the term is a perfect cube
therefore
The expression [tex]9w^{33} -y^{12}[/tex] is not a difference of cubes because the term [tex]9[/tex] is not a perfect cube
case B) [tex]18p^{15} -q^{21}[/tex]
we know that
[tex]18=2*3^{2}[/tex] ------> the term is not a perfect cube
[tex]p^{15}=(p^{5})^{3}[/tex] ------> the term is a perfect cube
[tex]q^{21}=(q^{7})^{3}[/tex] ------> the term is a perfect cube
therefore
The expression [tex]18p^{15} -q^{21}[/tex] is not a difference of cubes because the term [tex]18[/tex] is not a perfect cube
case C) [tex]36a^{22} -b^{16}[/tex]
we know that
[tex]36=2^{2}*3^{2}[/tex] ------> the term is not a perfect cube
[tex]a^{22}[/tex] ------> the term is not a perfect cube
[tex]b^{16}[/tex] ------> the term is not a perfect cube
therefore
The expression [tex]36a^{22} -b^{16}[/tex] is not a difference of cubes because all terms are not perfect cubes
case D) [tex]64c^{15} -a^{26}[/tex]
we know that
[tex]64=2^{6}=(2^{2})^{3}[/tex] ------> the term is a perfect cube
[tex]c^{15}=(c^{5})^{3}[/tex] ------> the term is a perfect cube
[tex]a^{26}[/tex] ------> the term is not a perfect cube
therefore
The expression [tex]64c^{15} -a^{26}[/tex] is not a difference of cubes because the term [tex]a^{26}[/tex] is not a perfect cube
I'm adding a new case so I can better explain the problem
case E) [tex]64c^{15} -d^{27}[/tex]
we know that
[tex]64=2^{6}=(2^{2})^{3}[/tex] ------> the term is a perfect cube
[tex]c^{15}=(c^{5})^{3}[/tex] ------> the term is a perfect cube
[tex]d^{27}=(d^{9})^{3}[/tex] ------> the term is a perfect cube
Substitute
[tex]64c^{15} -d^{27}=((2^{2})(c^{5}))^{3}-(d^{9})^{3}[/tex]
therefore
The expression [tex]64c^{15} -d^{27}[/tex] is a difference of cubes because all terms are perfect cubes
The expression [tex]\boxed{64{c^{15}} - {d^{27}}}[/tex] is a difference of cubes.
Further Explanation:
Given:
The options are as follows,
(a). [tex]9{w^{33}} - {y^{12}}[/tex]
(b). [tex]18{p^{15}} - {q^{21}}[/tex]
(c). [tex]36{a^{22}} - {b^{16}}[/tex]
(d). [tex]64{c^{15}} - {a^{26}}[/tex]
(e). [tex]64{c^{15}} - {d^{27}}[/tex]
Calculation:
The cubic formula can be expressed as follows,
[tex]\boxed{{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)}[/tex]
The expression is [tex]9{w^{33}} - {y^{12}}.[/tex]
9 is not a perfect cube of any number,[tex]{w^{33}}[/tex] can be written as [tex]{\left( {{w^{11}}} \right)^3}[/tex] and [tex]{y^{12}}[/tex] can be represents as [tex]{\left( {{y^4}} \right)^3}.[/tex]
[tex]9{w^{33}} - {y^{12}}[/tex]cannot be written as the difference of cube. Option (a) is not correct.
The expression is [tex]18{p^{15}} - {q^{21}}.[/tex]
18 is not a perfect cube of any number, [tex]{p^{15}}[/tex] can be written as [tex]{\left( {{p^5}} \right)^3}[/tex] and [tex]{q^{21}}[/tex] can be written as [tex]{\left( {{q^7}} \right)^3}.[/tex]
[tex]18{p^{15}} - {q^{21}}[/tex] cannot be written as the difference of cube. Option (b) is not correct.
The expression is [tex]36{a^{22}} - {b^{16}}.[/tex]
36 is not a perfect cube of any number, [tex]{a^{22}}[/tex] is not perfect cube and [tex]{b^{16}}[/tex] is not a perfect cube.
[tex]36{a^{22}} - {b^{16}}[/tex] cannot be written as the difference of cube. Option (c) is not correct.
The expression is [tex]64{c^{15}} - {a^{26}}.[/tex]
64 can be written as [tex]{\left( {{2^2}} \right)^3}, {a^{26}}[/tex] is not perfect cube and [tex]{c^{15}}[/tex] can be written as [tex]{\left( {{c^5}} \right)^3}.[/tex]
[tex]64{c^{15}} - {a^{26}}[/tex] cannot be written as the difference of cube. Option (d) is not correct.
The expression is [tex]64{c^{15}} - {d^{27}}.[/tex]
64 can be written as [tex]{\left( {{2^2}} \right)^3}, {d^{27}}[/tex] can be written as [tex]{\left( {{d^9}} \right)^3}[/tex] and [tex]{c^{15}}[/tex] can be written as [tex]{\left( {{c^5}} \right)^3}.[/tex]
[tex]\boxed{64{c^{15}} - {d^{27}} = {{\left( {{2^2}{c^5}} \right)}^3} - {{\left( {{d^9}} \right)}^3}}[/tex]
[tex]64{c^{15}} - {d^{27}}[/tex] can be written as the difference of cube. Option (e) is correct.
The expression [tex]\boxed{64{c^{15}} - {d^{27}}}[/tex] is a difference of cubes.
Learn more:
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Exponents and Powers
Keywords: Solution, factorized form, expression, difference of cubes, exponents, power, equation, power rule, exponent rule.
