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Which statement is true?

A. [tex]3^{-4} \ \textless \ \left(\frac{5}{6}\right)^2[/tex]
B. [tex]3^{-4} \ \textgreater \ \left(\frac{5}{6}\right)^3[/tex]
C. [tex]3^{-4} = \left(\frac{5}{6}\right)^3[/tex]

Sagot :

GinnyAnswer
Let's evaluate each statement step-by-step:

1. Compute [tex]\(3^{-4}\)[/tex]:
[tex]\[ 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \approx 0.012345679012345678 \][/tex]

2. Compute [tex]\(\left(\frac{5}{6}\right)^2\)[/tex]:
[tex]\[ \left(\frac{5}{6}\right)^2 = \frac{25}{36} \approx 0.6944444444444445 \][/tex]

3. Compute [tex]\(\left(\frac{5}{6}\right)^3\)[/tex]:
[tex]\[ \left(\frac{5}{6}\right)^3 = \frac{125}{216} \approx 0.5787037037037038 \][/tex]

Now let's compare the values to check each statement:

Statement (A):
[tex]\[ 3^{-4} < \left(\frac{5}{6}\right)^2 \][/tex]
This statement is comparing [tex]\(0.012345679012345678\)[/tex] and [tex]\(0.6944444444444445\)[/tex]. Since [tex]\(0.012345679012345678 < 0.6944444444444445\)[/tex], this statement is true.

Statement (B):
[tex]\[ 3^{-4} > \left(\frac{5}{6}\right)^3 \][/tex]
This statement is comparing [tex]\(0.012345679012345678\)[/tex] and [tex]\(0.5787037037037038\)[/tex]. Since [tex]\(0.012345679012345678 < 0.5787037037037038\)[/tex], this statement is false.

Statement (C):
[tex]\[ 3^{-4} = \left(\frac{5}{6}\right)^3 \][/tex]
This statement is comparing [tex]\(0.012345679012345678\)[/tex] and [tex]\(0.5787037037037038\)[/tex] once more. Since these two values are not equal, this statement is false.

Hence, the true statement is (A).
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