Question
Choose all the expressions that are equivalent to 4 5/8 (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 410−−−√5 5th root of 4 to the 10th power 45−−√8 8th root of 4 to the 5th power 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
Answers
Answered by
GPT-4o mini
To determine which expressions are equivalent to \(4 \frac{5}{8}\), we first convert \(4 \frac{5}{8}\) to an improper fraction:
\[
4 \frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{32 + 5}{8} = \frac{37}{8}.
\]
Next, we'll analyze each of the provided expressions to see which ones simplify to \(\frac{37}{8}\).
1. **Expression 1**: \( (478)(4^{-14}) \).
This expression is not in an easily comparable form, and without further simplification, we cannot directly relate it to \(4 \frac{5}{8}\).
2. **Expression 2**: \( (4^{7/8})(4^{-1/4}) \).
We can apply the properties of exponents:
\[
4^{7/8} \cdot 4^{-1/4} = 4^{7/8 - 1/4} = 4^{7/8 - 2/8} = 4^{5/8}.
\]
This does not equal \(4 \frac{5}{8}\).
3. **Expression 3**: \( \sqrt[5]{4^{10}} \).
This can be rewritten as:
\[
4^{10/5} = 4^2 = 16.
\]
This is not equal to \(4 \frac{5}{8}\).
4. **Expression 4**: \( \sqrt[8]{4^{5}} \).
This simplifies to:
\[
4^{5/8}.
\]
Again, this does not equal \(4 \frac{5}{8}\).
5. **Expression 5**: \( 4(4^{3/8}) \).
This simplifies to:
\[
4^{1 + 3/8} = 4^{8/8 + 3/8} = 4^{11/8}.
\]
This is not equal to \(4 \frac{5}{8}\).
6. **Expression 6**: \( (2148)(4^{-14}) \).
Similar to Expression 1, this will require further simplification that does not lead immediately to \(\frac{37}{8}\).
7. **Expression 7**: \( (2^{14/8})(4^{-1/4}) \).
This can be rewritten since \(4 = 2^2\):
\[
2^{14/8} \cdot 4^{-1/4} = 2^{14/8} \cdot 2^{-1/2} = 2^{14/8 - 2/8} = 2^{12/8} = 2^{3/2} = \sqrt{8}.
\]
This is not equivalent to \(4 \frac{5}{8}\).
After evaluating all expressions, none Simplify to \(4 \frac{5}{8}\) or \(\frac{37}{8}\).
Thus, none of the expressions provided are equivalent to \(4 \frac{5}{8}\).
\[
4 \frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{32 + 5}{8} = \frac{37}{8}.
\]
Next, we'll analyze each of the provided expressions to see which ones simplify to \(\frac{37}{8}\).
1. **Expression 1**: \( (478)(4^{-14}) \).
This expression is not in an easily comparable form, and without further simplification, we cannot directly relate it to \(4 \frac{5}{8}\).
2. **Expression 2**: \( (4^{7/8})(4^{-1/4}) \).
We can apply the properties of exponents:
\[
4^{7/8} \cdot 4^{-1/4} = 4^{7/8 - 1/4} = 4^{7/8 - 2/8} = 4^{5/8}.
\]
This does not equal \(4 \frac{5}{8}\).
3. **Expression 3**: \( \sqrt[5]{4^{10}} \).
This can be rewritten as:
\[
4^{10/5} = 4^2 = 16.
\]
This is not equal to \(4 \frac{5}{8}\).
4. **Expression 4**: \( \sqrt[8]{4^{5}} \).
This simplifies to:
\[
4^{5/8}.
\]
Again, this does not equal \(4 \frac{5}{8}\).
5. **Expression 5**: \( 4(4^{3/8}) \).
This simplifies to:
\[
4^{1 + 3/8} = 4^{8/8 + 3/8} = 4^{11/8}.
\]
This is not equal to \(4 \frac{5}{8}\).
6. **Expression 6**: \( (2148)(4^{-14}) \).
Similar to Expression 1, this will require further simplification that does not lead immediately to \(\frac{37}{8}\).
7. **Expression 7**: \( (2^{14/8})(4^{-1/4}) \).
This can be rewritten since \(4 = 2^2\):
\[
2^{14/8} \cdot 4^{-1/4} = 2^{14/8} \cdot 2^{-1/2} = 2^{14/8 - 2/8} = 2^{12/8} = 2^{3/2} = \sqrt{8}.
\]
This is not equivalent to \(4 \frac{5}{8}\).
After evaluating all expressions, none Simplify to \(4 \frac{5}{8}\) or \(\frac{37}{8}\).
Thus, none of the expressions provided are equivalent to \(4 \frac{5}{8}\).
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