Question 15

Question

Use Efficient Methods Without evaluating, order 5−7
, 54
, and 50
from least to greatest. Explain your reasoning. Choose the correct answer and explanation.

Multiple choice question.
cross out

A)
54, 50, 5−7
; The expression 54
is greater than 1, 50
equals 1, and when written with a positive exponent, 5−7
is 157
and is less than 1.

cross out

B)
5−7, 50,54
; The expression 5−7
has a negative value and is less than 1, 50
equals 0, and 54
is greater than 1.

cross out

C)
50,5−7, 54
; The expression 50
equals 0, written with a positive exponent, 5−7
is 157
and is less than 1, and 54
is greater than 1.

cross out

D)
5−7, 50,54
; Written with a positive exponent, 5−7
is 157
and is less than 1, 50
equals 1, and 54
is greater than 1.

Answer 1

To compare \( 5^{-7} \), \( 50 \), and \( 54 \) without evaluating them directly, let's analyze each expression.

1. Evaluate \( 5^{-7} \):

   • The expression \( 5^{-7} \) is equivalent to \( \frac{1}{5^7} \). Since \( 5^7 \) is a positive number greater than 1, \( \frac{1}{5^7} \) is a positive number less than 1.

2. Evaluate \( 50 \):

   • The expression \( 50 \) is equal to 1 because \( 5^0 = 1 \).

3. Evaluate \( 54 \):

   • The expression \( 54 \) is greater than 1 since \( 5^4 = 625 \) which is much greater than 1.

Now, we can order these three values:

• \( 5^{-7} < 1\) (since it is less than 1),
• \( 50 = 1 \),
• \( 54 > 1 \).

So, the correct order from least to greatest is:

• \( 5^{-7} < 50 < 54 \).

Now let’s look at the options provided:

A) 54, 50, 5−7 – Incorrect

B) 5−7, 50, 54 – Correct based on our analysis, but misidentified as negative in the explanation.

C) 50, 5−7, 54 – Incorrect

D) 5−7, 50, 54 – Correct based on our analysis and contains the right reasoning as well.

The best answer considering the reasoning offered, is: D) 5−7, 50, 54.