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To simplify and solve the given equation [tex]\( a^3 r^3 = -64 \)[/tex] for the variable [tex]\( a \)[/tex], we can follow these steps:
1. Observe the Equation:
The initial equation given is:
[tex]\[ a^3 r^3 = -64 \][/tex]
2. Assume a Value for [tex]\( r \)[/tex]:
Since there is no specific value provided for [tex]\( r \)[/tex], we will assume [tex]\( r = 1 \)[/tex]. This simplifies the equation significantly. By substituting [tex]\( r = 1 \)[/tex] into the equation, we get:
[tex]\[ a^3 \cdot 1^3 = -64 \][/tex]
This simplifies to:
[tex]\[ a^3 = -64 \][/tex]
3. Solve for [tex]\( a \)[/tex]:
To find [tex]\( a \)[/tex], we need to take the cube root of both sides of the equation [tex]\( a^3 = -64 \)[/tex].
[tex]\[ a = \sqrt[3]{-64} \][/tex]
4. Determine the Cube Root:
The cube root of [tex]\(-64\)[/tex] in the set of complex numbers can be determined. Recall that the cube roots of a number [tex]\( n \)[/tex] can be found using the formula for the roots of unity:
[tex]\[ \sqrt[3]{n} = \sqrt[3]{r e^{i(\theta + 2k\pi)/3}} \][/tex]
where [tex]\( r \)[/tex] is the magnitude and [tex]\( \theta \)[/tex] is the argument (angle) of the complex number, and [tex]\( k \)[/tex] denotes the different cube roots (typically [tex]\( k = 0, 1, 2 \)[/tex]).
For [tex]\(-64\)[/tex], we have:
[tex]\[ -64 = 64 e^{i\pi} \][/tex]
Thus, the principal cube root (when [tex]\( k = 0 \)[/tex]) is:
[tex]\[ \sqrt[3]{-64} = \sqrt[3]{64 e^{i \pi}} = \sqrt[3]{64} \cdot e^{i \pi / 3} \][/tex]
5. Simplify the Result:
The cube root of [tex]\( 64 \)[/tex] is [tex]\( 4 \)[/tex], and we account for the complex angle:
[tex]\[ a = 4 e^{i \pi / 3} \][/tex]
Converting [tex]\( e^{i \pi / 3} \)[/tex] back to Cartesian form yields:
[tex]\[ e^{i \pi / 3} = \cos(\pi / 3) + i \sin(\pi / 3) = \frac{1}{2} + i \frac{\sqrt{3}}{2} \][/tex]
Hence, we have:
[tex]\[ a = 4 \left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 2 + 2 \sqrt{3} i \][/tex]
Thus, [tex]\( a \)[/tex] is simplified to the complex number:
[tex]\[ a = 2 + 2 \sqrt{3} i \][/tex]
This means that:
[tex]\[ a^3 r^3 = -64 \][/tex]
simplifies such that:
[tex]\[ a = (2 + 2 \sqrt{3} i) \][/tex]
in the specified context of complex numbers.
Therefore, the complete solution gives [tex]\( a = (2 + 3.464101615137754i) \)[/tex].
1. Observe the Equation:
The initial equation given is:
[tex]\[ a^3 r^3 = -64 \][/tex]
2. Assume a Value for [tex]\( r \)[/tex]:
Since there is no specific value provided for [tex]\( r \)[/tex], we will assume [tex]\( r = 1 \)[/tex]. This simplifies the equation significantly. By substituting [tex]\( r = 1 \)[/tex] into the equation, we get:
[tex]\[ a^3 \cdot 1^3 = -64 \][/tex]
This simplifies to:
[tex]\[ a^3 = -64 \][/tex]
3. Solve for [tex]\( a \)[/tex]:
To find [tex]\( a \)[/tex], we need to take the cube root of both sides of the equation [tex]\( a^3 = -64 \)[/tex].
[tex]\[ a = \sqrt[3]{-64} \][/tex]
4. Determine the Cube Root:
The cube root of [tex]\(-64\)[/tex] in the set of complex numbers can be determined. Recall that the cube roots of a number [tex]\( n \)[/tex] can be found using the formula for the roots of unity:
[tex]\[ \sqrt[3]{n} = \sqrt[3]{r e^{i(\theta + 2k\pi)/3}} \][/tex]
where [tex]\( r \)[/tex] is the magnitude and [tex]\( \theta \)[/tex] is the argument (angle) of the complex number, and [tex]\( k \)[/tex] denotes the different cube roots (typically [tex]\( k = 0, 1, 2 \)[/tex]).
For [tex]\(-64\)[/tex], we have:
[tex]\[ -64 = 64 e^{i\pi} \][/tex]
Thus, the principal cube root (when [tex]\( k = 0 \)[/tex]) is:
[tex]\[ \sqrt[3]{-64} = \sqrt[3]{64 e^{i \pi}} = \sqrt[3]{64} \cdot e^{i \pi / 3} \][/tex]
5. Simplify the Result:
The cube root of [tex]\( 64 \)[/tex] is [tex]\( 4 \)[/tex], and we account for the complex angle:
[tex]\[ a = 4 e^{i \pi / 3} \][/tex]
Converting [tex]\( e^{i \pi / 3} \)[/tex] back to Cartesian form yields:
[tex]\[ e^{i \pi / 3} = \cos(\pi / 3) + i \sin(\pi / 3) = \frac{1}{2} + i \frac{\sqrt{3}}{2} \][/tex]
Hence, we have:
[tex]\[ a = 4 \left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 2 + 2 \sqrt{3} i \][/tex]
Thus, [tex]\( a \)[/tex] is simplified to the complex number:
[tex]\[ a = 2 + 2 \sqrt{3} i \][/tex]
This means that:
[tex]\[ a^3 r^3 = -64 \][/tex]
simplifies such that:
[tex]\[ a = (2 + 2 \sqrt{3} i) \][/tex]
in the specified context of complex numbers.
Therefore, the complete solution gives [tex]\( a = (2 + 3.464101615137754i) \)[/tex].
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