Answer :
To determine which choices are equivalent to the expression [tex]\(4 \sqrt{3}\)[/tex], we would need specific options to compare against it. However, I can explain how we can identify equivalent expressions for [tex]\(4 \sqrt{3}\)[/tex].
### Step-by-Step Explanation:
1. Understanding the Expression:
- The expression [tex]\(4 \sqrt{3}\)[/tex] refers to 4 times the square root of 3.
2. Equivalent Expressions:
- An equivalent expression is one that evaluates to the same number. For [tex]\(4 \sqrt{3}\)[/tex], this could include expressions that undergo transformation but remain mathematically the same in value.
3. Transformations:
- Multiplying by One (or Equivalent Forms): Expressions like [tex]\((2 \times 2) \sqrt{3}\)[/tex], [tex]\((\sqrt{12}) \times 2\)[/tex], etc., are equivalent because they ultimately simplify to [tex]\(4 \sqrt{3}\)[/tex].
- Expressing using Rational Forms: For example, [tex]\(\frac{4\sqrt{3} \times \sqrt{3}}{\sqrt{3}}\)[/tex] simplifies back to [tex]\(4 \sqrt{3}\)[/tex].
4. Simplifying and Rethinking Fractions/Roots:
- Any expression that reduces back to [tex]\(4 \sqrt{3}\)[/tex] through legitimate algebraic steps (e.g., rationalizing the denominator) is considered equivalent.
5. Numerical Calculation:
- If transformed expressions are difficult to simplify, calculate the numerical result of the transformation to verify equivalency against [tex]\(4 \sqrt{3}\)[/tex].
6. Substantial Transformations:
- Transformations such as [tex]\((\sqrt{48})\)[/tex], as [tex]\(\sqrt{48}\)[/tex] simplifies to [tex]\(4 \sqrt{3}\)[/tex], would also be an equivalent expression.
### Conclusion:
Without specific choices provided, identifying which are equivalent can be a bit open-ended. However, if you had multiple-choice options, applying these principles will help you determine which among those are actually equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Remember, for any expression to be equivalent to [tex]\(4 \sqrt{3}\)[/tex], it should translate back to the value of [tex]\(4 \times 1.732\ldots\)[/tex] roughly [tex]\(6.928\ldots\)[/tex], without deviation. If you can provide specific options, I’d be happy to confirm their equivalence!
### Step-by-Step Explanation:
1. Understanding the Expression:
- The expression [tex]\(4 \sqrt{3}\)[/tex] refers to 4 times the square root of 3.
2. Equivalent Expressions:
- An equivalent expression is one that evaluates to the same number. For [tex]\(4 \sqrt{3}\)[/tex], this could include expressions that undergo transformation but remain mathematically the same in value.
3. Transformations:
- Multiplying by One (or Equivalent Forms): Expressions like [tex]\((2 \times 2) \sqrt{3}\)[/tex], [tex]\((\sqrt{12}) \times 2\)[/tex], etc., are equivalent because they ultimately simplify to [tex]\(4 \sqrt{3}\)[/tex].
- Expressing using Rational Forms: For example, [tex]\(\frac{4\sqrt{3} \times \sqrt{3}}{\sqrt{3}}\)[/tex] simplifies back to [tex]\(4 \sqrt{3}\)[/tex].
4. Simplifying and Rethinking Fractions/Roots:
- Any expression that reduces back to [tex]\(4 \sqrt{3}\)[/tex] through legitimate algebraic steps (e.g., rationalizing the denominator) is considered equivalent.
5. Numerical Calculation:
- If transformed expressions are difficult to simplify, calculate the numerical result of the transformation to verify equivalency against [tex]\(4 \sqrt{3}\)[/tex].
6. Substantial Transformations:
- Transformations such as [tex]\((\sqrt{48})\)[/tex], as [tex]\(\sqrt{48}\)[/tex] simplifies to [tex]\(4 \sqrt{3}\)[/tex], would also be an equivalent expression.
### Conclusion:
Without specific choices provided, identifying which are equivalent can be a bit open-ended. However, if you had multiple-choice options, applying these principles will help you determine which among those are actually equivalent to [tex]\(4 \sqrt{3}\)[/tex].
Remember, for any expression to be equivalent to [tex]\(4 \sqrt{3}\)[/tex], it should translate back to the value of [tex]\(4 \times 1.732\ldots\)[/tex] roughly [tex]\(6.928\ldots\)[/tex], without deviation. If you can provide specific options, I’d be happy to confirm their equivalence!