Respuesta :
Finding the cube root of each expression
Expression A:
[tex] \sqrt[3]{-8x^{21}y^{8}} [/tex]
[tex] \sqrt[3]{-8} [/tex] [tex] \sqrt[3]{x^{21}[tex] \sqrt[3]{y^8} [/tex]} [/tex]
[tex]-2x^ \frac{21}{3} y^ \frac{8}{3} [/tex]
[tex]-2 x^7 y^ \frac{8}{3} [/tex]
the term [tex]y^ \frac{8}{3} [/tex] is not a perfect cube, therefore expression A is not a perfect cube.
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Expression B
[tex] \sqrt[3]{-64x^{64}y^{64}} [/tex]
[tex] \sqrt[3]{-64} \sqrt[3]{x^{64}} \sqrt[3]{y^{64}} [/tex]
[tex]-4 x^ \frac{64}{3} y^ \frac{64}{3} [/tex]
The term [tex]x^ \frac{64}{3} [/tex] and [tex]y^ \frac{64}{3} [/tex] are not perfect cube because 64/3 doesn't give whole number
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Expression C
[tex] \sqrt[3] {-125x^9y^{20}} [/tex]
[tex] \sqrt[3]{-125} \sqrt[3]{x^9} \sqrt[3] {y^{20}} [/tex]
[tex]-5 (x^ \frac{9}{3}) (y^ \frac{20}{3})[/tex]
[tex]-5 (x^3) (y^ \frac{20} {3}) [/tex]
The term [tex]y^ \frac{20}{3} [/tex] are not perfect cube because 20/3 doesn't give whole number
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Expression D
[tex] \sqrt[3]{ -216 x^{9} y^{18} } [/tex]
[tex] \sqrt[3]{-216} \sqrt[3]{x^9} \sqrt[3]{y^{18}} [/tex]
[tex]-6 ( x^ \frac{9} {3} ) ( y^ \frac{18} {3} ) [/tex]
[tex]-6 (x^3) (y^6)[/tex]
All terms are perfect cubes
ANSWER: OPTION D
Expression A:
[tex] \sqrt[3]{-8x^{21}y^{8}} [/tex]
[tex] \sqrt[3]{-8} [/tex] [tex] \sqrt[3]{x^{21}[tex] \sqrt[3]{y^8} [/tex]} [/tex]
[tex]-2x^ \frac{21}{3} y^ \frac{8}{3} [/tex]
[tex]-2 x^7 y^ \frac{8}{3} [/tex]
the term [tex]y^ \frac{8}{3} [/tex] is not a perfect cube, therefore expression A is not a perfect cube.
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Expression B
[tex] \sqrt[3]{-64x^{64}y^{64}} [/tex]
[tex] \sqrt[3]{-64} \sqrt[3]{x^{64}} \sqrt[3]{y^{64}} [/tex]
[tex]-4 x^ \frac{64}{3} y^ \frac{64}{3} [/tex]
The term [tex]x^ \frac{64}{3} [/tex] and [tex]y^ \frac{64}{3} [/tex] are not perfect cube because 64/3 doesn't give whole number
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Expression C
[tex] \sqrt[3] {-125x^9y^{20}} [/tex]
[tex] \sqrt[3]{-125} \sqrt[3]{x^9} \sqrt[3] {y^{20}} [/tex]
[tex]-5 (x^ \frac{9}{3}) (y^ \frac{20}{3})[/tex]
[tex]-5 (x^3) (y^ \frac{20} {3}) [/tex]
The term [tex]y^ \frac{20}{3} [/tex] are not perfect cube because 20/3 doesn't give whole number
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Expression D
[tex] \sqrt[3]{ -216 x^{9} y^{18} } [/tex]
[tex] \sqrt[3]{-216} \sqrt[3]{x^9} \sqrt[3]{y^{18}} [/tex]
[tex]-6 ( x^ \frac{9} {3} ) ( y^ \frac{18} {3} ) [/tex]
[tex]-6 (x^3) (y^6)[/tex]
All terms are perfect cubes
ANSWER: OPTION D
AThe expression [tex]\boxed{ - 125{x^9}{y^{18}}}[/tex] is a perfect cube.
Further Explanation:
Given:
The options are as follows,
(a). [tex]- 8{x^{21}}{y^8}[/tex]
(b). [tex]- 64{x^{64}}{y^{64}}[/tex]
(c). [tex]- 125{x^9}{y^{20}}[/tex]
(d). [tex]- 125{x^9}{y^{18}}[/tex]
Calculation:
The expression is [tex]- 8{x^{21}}{y^8}.[/tex]
Solve the above expression.
[tex]\begin{aligned}A&= \sqrt[3]{{ - 8{x^{21}}{y^8}}}\\&= \sqrt[3]{{{{\left( { - 2} \right)}^3}{{\left( {{x^7}} \right)}^3}{y^8}}} \\ &= - 2{x^7}{y^{\frac{8}{3}}}\\\end{aligned}[/tex]
[tex]- 8{x^{21}}{y^8}[/tex] cannot be written as the perfect cube. Option (a) is not correct.
The expression is [tex]- 64{x^{64}}{y^{64}}.[/tex]
Solve the above expression.
[tex]\begin{aligned}B&= \sqrt[3]{{ - 64{x^{64}}{y^{64}}}}\\&= \sqrt[3]{{{{\left( { - 4} \right)}^3}{x^{64}}{y^{64}}}}\\&= - 4{x^{\frac{{64}}{3}}}{y^{\frac{{64}}{3}}}\\\end{aligned}[/tex]
[tex]- 64{x^{64}}{y^{64}}[/tex] cannot be written as the perfect cube. Option (b) is not correct.
The expression is [tex]- 125{x^9}{y^{20}}.[/tex]
Solve the above expression.
[tex]\begin{aligned}C &= \sqrt[3]{{ - 125{x^9}{y^{20}}}}\\&= \sqrt[3]{{{{\left( { - 5} \right)}^3}{{\left( {{x^3}} \right)}^3}{y^{20}}}} \\&= - 5{x^3}{y^{\frac{{20}}{3}}}\\\end{aligned}[/tex]
[tex]- 125{x^9}{y^{20}}[/tex] cannot be written as the perfect cube. Option (c) is not correct.
The expression is [tex]- 125{x^9}{y^{18}}.[/tex]
Solve the above expression.
[tex]\begin{aligned}D&= \sqrt[3]{{ - 125{x^9}{y^{18}}}}\\&= \sqrt[3]{{{{\left( { - 5} \right)}^3}{{\left( {{x^3}} \right)}^3}{{\left( {{y^6}} \right)}^3}}} \\&=- 5{x^3}{y^6}\\\end{aligned}[/tex]
[tex]- 125{x^9}{y^{18}}[/tex] can be written as the perfect cube. Option (d) is correct.
The expression [tex]\boxed{ - 125{x^9}{y^{18}}}[/tex] is a perfect cube.
Learn more:
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Exponents and Powers
Keywords: Solution, perfect cube, factorized form, expression, difference of cubes, exponents, power, equation, power rule, exponent rule.
