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Question

Evaluate each expression. Then, list the expressions in order from greatest to least according to their values.

2^(-2) x 2^(-3) 7^8/7^8 4^2 x 4^(-1) 5^(-10)/5^(-12)

Answer 1

To evaluate the given expressions, we'll simplify them one by one.

1. Expression: \( 2^{-2} \times 2^{-3} \)

   • Using the property of exponents: \( a^m \times a^n = a^{m+n} \)
   • \( 2^{-2} \times 2^{-3} = 2^{-2 + (-3)} = 2^{-5} = \frac{1}{2^5} = \frac{1}{32} \)

2. Expression: \( \frac{7^8}{7^8} \)

   • Any non-zero number divided by itself equals 1.
   • \( \frac{7^8}{7^8} = 1 \)

3. Expression: \( 4^2 \times 4^{-1} \)

   • Using the property of exponents: \( a^m \times a^n = a^{m+n} \)
   • \( 4^2 \times 4^{-1} = 4^{2 + (-1)} = 4^{1} = 4 \)

4. Expression: \( \frac{5^{-10}}{5^{-12}} \)

   • Using the property of exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
   • \( \frac{5^{-10}}{5^{-12}} = 5^{-10 - (-12)} = 5^{2} = 25 \)

Now we have evaluated each expression:

1. \( 2^{-2} \times 2^{-3} = \frac{1}{32} \)
2. \( \frac{7^8}{7^8} = 1 \)
3. \( 4^2 \times 4^{-1} = 4 \)
4. \( \frac{5^{-10}}{5^{-12}} = 25 \)

Next, we will list the values in order from greatest to least:

• 25
• 4
• 1
• \(\frac{1}{32}\)

Now let's write them in order according to the expressions:

• \( \frac{5^{-10}}{5^{-12}} \) (25)
• \( 4^2 \times 4^{-1} \) (4)
• \( \frac{7^8}{7^8} \) (1)
• \( 2^{-2} \times 2^{-3} \) (\(\frac{1}{32}\))

So the final order from greatest to least is:

1. \( \frac{5^{-10}}{5^{-12}} \)
2. \( 4^2 \times 4^{-1} \)
3. \( \frac{7^8}{7^8} \)
4. \( 2^{-2} \times 2^{-3} \)