Question

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

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

(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

45−−√8
8th root of 4 to the 5th power

(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

Answer 1

To determine which expressions are equivalent to 458, let's simplify each of the responses one by one.

1. **410−−−√5** (Square root of 410):
\[
\sqrt{410} \approx 20.248 \quad (\text{not equivalent to } 458)
\]

2. **5th root of 4 to the 10th power**:
\[
\sqrt[5]{4^{10}} = 4^{10/5} = 4^2 = 16 \quad (\text{not equivalent to } 458)
\]

3. **(2148)(4−14)**:
First, calculate \( 4^{-14} \):
\[
4^{-14} = \frac{1}{4^{14}} \quad (\text{so this term will be very small})
\]
\[
2^{14} \times 4^{-14} = 2^{14} \times \left(\frac{1}{4^{14}}\right) = 2^{14} \times \left(\frac{1}{(2^2)^{14}}\right) = 2^{14} \times \frac{1}{2^{28}} = 2^{14 - 28} = 2^{-14} \quad (\text{not equivalent to } 458)
\]

4. **45−−√8** (Square root of 45):
\[
\sqrt{45} \approx 6.708 \quad (\text{not equivalent to } 458)
\]

5. **(478)(4−14)**:
Similar to before, we'll calculate:
\[
4^{7/8} \times 4^{-14} = 4^{7/8 - 14} = 4^{7/8 - 112/8} = 4^{-105/8} \quad (\text{very small; not equivalent to } 458)
\]

6. **4(438)**:
Simplify \( 4 \times 4^{3/8} = 4^{1 + 3/8} = 4^{8/8 + 3/8} = 4^{11/8} = (2^2)^{11/8} = 2^{11/4}\).
\[
2^{11/4} \quad (\text{This expression does not evaluate further to } 458.)
\]

After evaluating all expressions, none are equivalent to 458. Hence, the answer is **none of the expressions listed** are equivalent to 458.