Question

Choose all the expressions that are equivalent to 458
(1 point)
Responses

(478)(4−14)
open paren 4 raised to the 7 eighths power close paren times open paren 4 raised to the negative 1 fourth power close paren

4(438)
4 times open paren 4 raised to the 3 eighths power close paren

(2148)(4−14)
open paren 2 raised to the 14 eighths power close paren times open paren 4 raised to the negative 1 fourth power close paren

410−−−√5
5th root of 4 to the 10th power

45−−√8
8th root of 4 to the 5th power

Answer 1

To determine which expressions are equivalent to 458, we need to simplify or evaluate each expression:

1. **(478)(4−14)**:
This can be rewritten as:
\( 4^{7/8} \cdot 4^{-1/4} = 4^{7/8 - 1/4} = 4^{7/8 - 2/8} = 4^{5/8} \).
We need to compare this with \( 4^{5/8} \). Since \( 4^{5/8} \) is not equal to 458, this expression is **not equivalent** to 458.

2. **4(438)**:
This can be rewritten as:
\( 4 \cdot 4^{3/8} = 4^{1 + 3/8} = 4^{8/8 + 3/8} = 4^{11/8} \).
We find that \( 4^{11/8} \) is also not equal to 458, so this expression is **not equivalent** to 458.

3. **(2148)(4−14)**:
This can be rewritten as:
\( 2^{14/8} \cdot 4^{-1/4} = 2^{14/8} \cdot 4^{-2/8} = 2^{14/8} \cdot 2^{-1/4 \cdot 2} = 2^{14/8 - 2/8} = 2^{12/8} = 2^{3/2} = \sqrt{8} \).
Calculating \( \sqrt{8} \) gives us \( 2\sqrt{2} \approx 2(1.414) = 2.828 \), which is not equal to 458. Hence, this expression is **not equivalent** to 458.

4. **410−−−√5**:
This can be rewritten as:
\( \sqrt[5]{4^{10}} = (4^2)^{2/5} = 16^{2/5} \).
Computing \( 16^{2/5} = (2^4)^{2/5} = 2^{8/5} \). Evaluating this would yield approximately \( 2^{1.6} \approx 3.17 \), which is not equal to 458. Therefore, this expression is **not equivalent** to 458.

5. **45−−√8**:
This can be rewritten as:
\( \sqrt[8]{4^5} = (2^2)^{5/8} = 2^{10/8} = 2^{5/4} = \sqrt[4]{32} \).
Estimating this gives us approximately \( 3.17 \), which is not equal to 458, so this expression is **not equivalent** to 458.

In conclusion, **none** of the given expressions are equivalent to 458.