Which expression is a difference of cubes?
X^6-6
X^6-8
x^8-6
X^8-8
Answer: The correct expression is, \(x^6-8\)
Step-by-step explanation:
\(x^6\) is represented in cube form as, \((x^2)^3\)
'8' is represented in cube form as, \(2^3\)
\(x^8\) and '6' will not show cube form of integer power.
The expanded form of the given expression, \(x^6-8\) is represented as,
\(x^6-8=(x^2)^3-2^3\)
This expression will showing the difference of cubes.
And the other options, \(x^6-6,x^8-6,x^8-8\) will not show the difference of cubes.
Therefore, the correct answer is, \(x^6-8\)
The expression \(\boxed{{x^6} - 8}\) is a difference of cubes. Option (b) is correct.
Further Explanation:
The cubic formula can be expressed as follows,
\(\boxed{{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)}\)
Given:
The options are as follows,
(a). \({x^6} - 6\)
(b). \({x^6} - 8\)
(c). \({x^8} - 6\)
(d). \({x^8} - 8\)
Calculation:
8 is a cube of 2 and can be written as follows,
\(8 = {2^3}\)
\({x^6}\) can be written as a cube of \({x^2}.\)
\({x^6} = {\left( {{x^2}} \right)^3}\)
Use the identity \({a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\) in above expression.
\(\begin{aligned}{x^6} - 8&= {\left( {{x^2}} \right)^3}- {\left( 2 \right)^3} \\&= \left( {{x^2} - 2} \right)\left( {{x^4} + 2{x^2} + 4} \right)\\\end{aligned}\)
The expression \(\boxed{{x^6} - 8}\) is a difference of cubes. Option (b) is correct.
Option (a) is not correct.
Option (b) is correct.
Option (c) is not correct.
Option (d) is not correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Exponents and Powers
Keywords: Solution, factorized form, \(x^12y^18+1\), exponents, power, equation, power rule, exponent rule.