Verified Answer :
we know that
A polynomial in the form \(a^{3}-b^{3}\) 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) \(9w^{33} -y^{12}\)
we know that
\(9=3^{2}\) ------> the term is not a perfect cube
\(w^{33}=(w^{11})^{3}\) ------> the term is a perfect cube
\(y^{12}=(y^{4})^{3}\) ------> the term is a perfect cube
therefore
The expression \(9w^{33} -y^{12}\) is not a difference of cubes because the term \(9\) is not a perfect cube
case B) \(18p^{15} -q^{21}\)
we know that
\(18=2*3^{2}\) ------> the term is not a perfect cube
\(p^{15}=(p^{5})^{3}\) ------> the term is a perfect cube
\(q^{21}=(q^{7})^{3}\) ------> the term is a perfect cube
therefore
The expression \(18p^{15} -q^{21}\) is not a difference of cubes because the term \(18\) is not a perfect cube
case C) \(36a^{22} -b^{16}\)
we know that
\(36=2^{2}*3^{2}\) ------> the term is not a perfect cube
\(a^{22}\) ------> the term is not a perfect cube
\(b^{16}\) ------> the term is not a perfect cube
therefore
The expression \(36a^{22} -b^{16}\) is not a difference of cubes because all terms are not perfect cubes
case D) \(64c^{15} -a^{26}\)
we know that
\(64=2^{6}=(2^{2})^{3}\) ------> the term is a perfect cube
\(c^{15}=(c^{5})^{3}\) ------> the term is a perfect cube
\(a^{26}\) ------> the term is not a perfect cube
therefore
The expression \(64c^{15} -a^{26}\) is not a difference of cubes because the term \(a^{26}\) is not a perfect cube
I'm adding a new case so I can better explain the problem
case E) \(64c^{15} -d^{27}\)
we know that
\(64=2^{6}=(2^{2})^{3}\) ------> the term is a perfect cube
\(c^{15}=(c^{5})^{3}\) ------> the term is a perfect cube
\(d^{27}=(d^{9})^{3}\) ------> the term is a perfect cube
Substitute
\(64c^{15} -d^{27}=((2^{2})(c^{5}))^{3}-(d^{9})^{3}\)
therefore
The expression \(64c^{15} -d^{27}\) is a difference of cubes because all terms are perfect cubes
The expression \(\boxed{64{c^{15}} - {d^{27}}}\) is a difference of cubes.
Further Explanation:
Given:
The options are as follows,
(a). \(9{w^{33}} - {y^{12}}\)
(b). \(18{p^{15}} - {q^{21}}\)
(c). \(36{a^{22}} - {b^{16}}\)
(d). \(64{c^{15}} - {a^{26}}\)
(e). \(64{c^{15}} - {d^{27}}\)
Calculation:
The cubic formula can be expressed as follows,
\(\boxed{{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)}\)
The expression is \(9{w^{33}} - {y^{12}}.\)
9 is not a perfect cube of any number,\({w^{33}}\) can be written as \({\left( {{w^{11}}} \right)^3}\) and \({y^{12}}\) can be represents as \({\left( {{y^4}} \right)^3}.\)
\(9{w^{33}} - {y^{12}}\)cannot be written as the difference of cube. Option (a) is not correct.
The expression is \(18{p^{15}} - {q^{21}}.\)
18 is not a perfect cube of any number, \({p^{15}}\) can be written as \({\left( {{p^5}} \right)^3}\) and \({q^{21}}\) can be written as \({\left( {{q^7}} \right)^3}.\)
\(18{p^{15}} - {q^{21}}\) cannot be written as the difference of cube. Option (b) is not correct.
The expression is \(36{a^{22}} - {b^{16}}.\)
36 is not a perfect cube of any number, \({a^{22}}\) is not perfect cube and \({b^{16}}\) is not a perfect cube.
\(36{a^{22}} - {b^{16}}\) cannot be written as the difference of cube. Option (c) is not correct.
The expression is \(64{c^{15}} - {a^{26}}.\)
64 can be written as \({\left( {{2^2}} \right)^3}, {a^{26}}\) is not perfect cube and \({c^{15}}\) can be written as \({\left( {{c^5}} \right)^3}.\)
\(64{c^{15}} - {a^{26}}\) cannot be written as the difference of cube. Option (d) is not correct.
The expression is \(64{c^{15}} - {d^{27}}.\)
64 can be written as \({\left( {{2^2}} \right)^3}, {d^{27}}\) can be written as \({\left( {{d^9}} \right)^3}\) and \({c^{15}}\) can be written as \({\left( {{c^5}} \right)^3}.\)
\(\boxed{64{c^{15}} - {d^{27}} = {{\left( {{2^2}{c^5}} \right)}^3} - {{\left( {{d^9}} \right)}^3}}\)
\(64{c^{15}} - {d^{27}}\) can be written as the difference of cube. Option (e) is correct.
The expression \(\boxed{64{c^{15}} - {d^{27}}}\) is a difference of cubes.
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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.