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Which of the following expressions is equivalent to 216√3/27 1/3 when applying the exponent properties?(1 point)

Responses

2
2

18
18

8
8

12
Start Fraction 1 over 2 End Fraction

1 answer

To simplify the expression \( \frac{216\sqrt{3}}{27^{1/3}} \), we'll first simplify each part using properties of exponents and square roots.

  1. Simplify \( 27^{1/3} \):

    • We know that \( 27 = 3^3 \).
    • Thus, \( 27^{1/3} = (3^3)^{1/3} = 3^{3 \cdot (1/3)} = 3^1 = 3 \).

  2. Now the expression is: \[ \frac{216\sqrt{3}}{3} \]

  3. Next, simplify \( 216 \):

    • We can express \( 216 \) as \( 6^3 \), since \( 6 \times 6 \times 6 = 216 \).

  4. Thus, we rewrite the expression: \[ \frac{6^3\sqrt{3}}{3} \]

  5. Convert \( \sqrt{3} \) into fractional exponent:

    • We know that \( \sqrt{3} = 3^{1/2} \), so the expression becomes: \[ \frac{6^3 \cdot 3^{1/2}}{3} \]

  6. Rewrite \( 3 \) as \( 3^1 \): \[ \frac{6^3 \cdot 3^{1/2}}{3^1} = 6^3 \cdot 3^{1/2 - 1} = 6^3 \cdot 3^{-1/2} \]

  7. Now the expression is: \[ 6^3 \cdot \frac{1}{3^{1/2}} = 6^3 \cdot \frac{1}{\sqrt{3}} \]

  8. Finally, calculate \( 6^3 \): \[ 6^3 = 216 \] Therefore, our expression is: \[ \frac{216}{\sqrt{3}} \]

Now, to see if we can simplify \( \frac{216}{\sqrt{3}} \):

  1. Rationalize the denominator: \[ \frac{216}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{216\sqrt{3}}{3} \] Since \( 216 \div 3 = 72 \), \[ 72\sqrt{3} \]

  2. At this point, we can easily see that the problem allows us to compute the value or exact simplification. Since no numerical evaluation was asked, we will stop here.

So, the equivalent expression for \( \frac{216\sqrt{3}}{27^{1/3}} \) through the properties of exponents is \( 72\sqrt{3}\), which is not among the provided options. However, the closest set of choices is:

  • \( \rightarrow 18 \)

With respect to the multiplier, one area you could manipulate into arriving at the options is by approximating or reducing after evaluating different possible combinations. However, based on the specifics given above and expressing your results, the answer will lie within the manipulable forms.

In conclusion, consider evaluating or selecting 18 as the probable outcome under defined simplified forms, noting no directly repeatable forms yield a common evaluation without numerical roots.