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To solve the equation [tex]\( m^3 = 8 \)[/tex], we need to determine the values of [tex]\( m \)[/tex] such that when [tex]\( m \)[/tex] is cubed, the result is 8. Let's go through the steps to find these solutions:
1. Identify the cube root of 8:
By definition, the cube root of a number [tex]\( n \)[/tex] is a value [tex]\( m \)[/tex] such that when raised to the power of 3, it equals [tex]\( n \)[/tex]. Thus, for [tex]\( m^3 = 8 \)[/tex], we need to find the value of [tex]\( m \)[/tex] such that:
[tex]\[ m = \sqrt[3]{8} \][/tex]
2. Solving the equation:
- Calculate [tex]\( \sqrt[3]{8} \)[/tex]:
We know that 8 is equal to [tex]\( 2^3 \)[/tex]. Consequently:
[tex]\[ \sqrt[3]{8} = \sqrt[3]{2^3} = 2 \][/tex]
Hence, [tex]\( m = 2 \)[/tex] is a solution.
- Confirm that [tex]\( m = 2 \)[/tex] satisfies the original equation:
[tex]\[ (2)^3 = 2 \times 2 \times 2 = 8 \][/tex]
This confirms that [tex]\( m = 2 \)[/tex] is indeed a solution.
3. Evaluate the other potential solutions given in the options:
- Option A: 2
As shown above, [tex]\( 2 \)[/tex] is indeed a solution because [tex]\( 2^3 = 8 \)[/tex].
- Option B: 4
[tex]\[ 4^3 = 4 \times 4 \times 4 = 64 \][/tex]
Hence, [tex]\( 4 \)[/tex] is not a solution.
- Option C: 243
[tex]\[ 243^3 = 243 \times 243 \times 243 \][/tex]
Clearly, this value is much larger than 8, so [tex]\( 243 \)[/tex] is not a solution.
- Option D: [tex]\( \sqrt[3]{8} \)[/tex]
By the calculation above, we have:
[tex]\[ \sqrt[3]{8} = 2 \][/tex]
So, [tex]\( \sqrt[3]{8} \)[/tex] is indeed a solution.
- Option E: [tex]\( \sqrt{8} \)[/tex]
[tex]\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \][/tex]
And,
[tex]\[ (2\sqrt{2})^3 = 8 \sqrt{2^3} = 8\sqrt{8} \][/tex]
Hence, [tex]\( \sqrt{8} \)[/tex] is not a solution.
Based on the detailed evaluation, the two values that are solutions to the equation [tex]\( m^3 = 8 \)[/tex] are:
- Option A: 2
- Option D: [tex]\( \sqrt[3]{8} \)[/tex]
Therefore, the correct choices are A (2) and D ([tex]\(\sqrt[3]{8}\)[/tex]).
1. Identify the cube root of 8:
By definition, the cube root of a number [tex]\( n \)[/tex] is a value [tex]\( m \)[/tex] such that when raised to the power of 3, it equals [tex]\( n \)[/tex]. Thus, for [tex]\( m^3 = 8 \)[/tex], we need to find the value of [tex]\( m \)[/tex] such that:
[tex]\[ m = \sqrt[3]{8} \][/tex]
2. Solving the equation:
- Calculate [tex]\( \sqrt[3]{8} \)[/tex]:
We know that 8 is equal to [tex]\( 2^3 \)[/tex]. Consequently:
[tex]\[ \sqrt[3]{8} = \sqrt[3]{2^3} = 2 \][/tex]
Hence, [tex]\( m = 2 \)[/tex] is a solution.
- Confirm that [tex]\( m = 2 \)[/tex] satisfies the original equation:
[tex]\[ (2)^3 = 2 \times 2 \times 2 = 8 \][/tex]
This confirms that [tex]\( m = 2 \)[/tex] is indeed a solution.
3. Evaluate the other potential solutions given in the options:
- Option A: 2
As shown above, [tex]\( 2 \)[/tex] is indeed a solution because [tex]\( 2^3 = 8 \)[/tex].
- Option B: 4
[tex]\[ 4^3 = 4 \times 4 \times 4 = 64 \][/tex]
Hence, [tex]\( 4 \)[/tex] is not a solution.
- Option C: 243
[tex]\[ 243^3 = 243 \times 243 \times 243 \][/tex]
Clearly, this value is much larger than 8, so [tex]\( 243 \)[/tex] is not a solution.
- Option D: [tex]\( \sqrt[3]{8} \)[/tex]
By the calculation above, we have:
[tex]\[ \sqrt[3]{8} = 2 \][/tex]
So, [tex]\( \sqrt[3]{8} \)[/tex] is indeed a solution.
- Option E: [tex]\( \sqrt{8} \)[/tex]
[tex]\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \][/tex]
And,
[tex]\[ (2\sqrt{2})^3 = 8 \sqrt{2^3} = 8\sqrt{8} \][/tex]
Hence, [tex]\( \sqrt{8} \)[/tex] is not a solution.
Based on the detailed evaluation, the two values that are solutions to the equation [tex]\( m^3 = 8 \)[/tex] are:
- Option A: 2
- Option D: [tex]\( \sqrt[3]{8} \)[/tex]
Therefore, the correct choices are A (2) and D ([tex]\(\sqrt[3]{8}\)[/tex]).
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