Answer :
To determine which expressions are equivalent to [tex]\( \left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3 \)[/tex], let’s follow the steps to simplify the given expression:
1. Combine the exponents inside the parentheses:
[tex]\[ 5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} \][/tex]
When multiplying two exponents with the same base, we add the exponents:
[tex]\[ 5^{\frac{1}{8} + \frac{3}{8}} = 5^{\frac{4}{8}} = 5^{\frac{1}{2}} \][/tex]
2. Raise the simplified expression to the power of 3:
[tex]\[ \left(5^{\frac{1}{2}}\right)^3 \][/tex]
When raising a power to a power, we multiply the exponents:
[tex]\[ 5^{\frac{1}{2} \cdot 3} = 5^{\frac{3}{2}} \][/tex]
Therefore, the expression [tex]\( \left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3 \)[/tex] simplifies to [tex]\( 5^{\frac{3}{2}} \)[/tex].
Let's evaluate the given expressions to see which ones are equivalent to [tex]\( 5^{\frac{3}{2}} \)[/tex]:
1. [tex]\( 5^{\frac{3}{2}} \)[/tex]:
- This is exactly the expression we obtained, so it is equivalent.
2. [tex]\( 5^{\frac{9}{8}} \)[/tex]:
- This has a different exponent ([tex]\(\frac{9}{8}\)[/tex]), so it is not equivalent.
3. [tex]\( \sqrt{5^3} \)[/tex]:
- Recall that the square root of a number is the same as raising it to the power of [tex]\(\frac{1}{2}\)[/tex]. Thus, this can be rewritten as:
[tex]\[ \sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{3 \cdot \frac{1}{2}} = 5^{\frac{3}{2}} \][/tex]
- This is equivalent to [tex]\( 5^{\frac{3}{2}} \)[/tex].
4. [tex]\( (\sqrt[8]{5})^9 \)[/tex]:
- This can be rewritten using fractional exponents. Recall that [tex]\(\sqrt[8]{5} = 5^{\frac{1}{8}}\)[/tex]. Therefore:
[tex]\[ (\sqrt[8]{5})^9 = \left(5^{\frac{1}{8}}\right)^9 = 5^{\frac{1}{8} \cdot 9} = 5^{\frac{9}{8}} \][/tex]
- This is not equivalent to [tex]\( 5^{\frac{3}{2}} \)[/tex].
The expressions that are equivalent to [tex]\( \left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3 \)[/tex] are:
- [tex]\( 5^{\frac{3}{2}} \)[/tex]
- [tex]\( \sqrt{5^3} \)[/tex]
1. Combine the exponents inside the parentheses:
[tex]\[ 5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}} \][/tex]
When multiplying two exponents with the same base, we add the exponents:
[tex]\[ 5^{\frac{1}{8} + \frac{3}{8}} = 5^{\frac{4}{8}} = 5^{\frac{1}{2}} \][/tex]
2. Raise the simplified expression to the power of 3:
[tex]\[ \left(5^{\frac{1}{2}}\right)^3 \][/tex]
When raising a power to a power, we multiply the exponents:
[tex]\[ 5^{\frac{1}{2} \cdot 3} = 5^{\frac{3}{2}} \][/tex]
Therefore, the expression [tex]\( \left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3 \)[/tex] simplifies to [tex]\( 5^{\frac{3}{2}} \)[/tex].
Let's evaluate the given expressions to see which ones are equivalent to [tex]\( 5^{\frac{3}{2}} \)[/tex]:
1. [tex]\( 5^{\frac{3}{2}} \)[/tex]:
- This is exactly the expression we obtained, so it is equivalent.
2. [tex]\( 5^{\frac{9}{8}} \)[/tex]:
- This has a different exponent ([tex]\(\frac{9}{8}\)[/tex]), so it is not equivalent.
3. [tex]\( \sqrt{5^3} \)[/tex]:
- Recall that the square root of a number is the same as raising it to the power of [tex]\(\frac{1}{2}\)[/tex]. Thus, this can be rewritten as:
[tex]\[ \sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{3 \cdot \frac{1}{2}} = 5^{\frac{3}{2}} \][/tex]
- This is equivalent to [tex]\( 5^{\frac{3}{2}} \)[/tex].
4. [tex]\( (\sqrt[8]{5})^9 \)[/tex]:
- This can be rewritten using fractional exponents. Recall that [tex]\(\sqrt[8]{5} = 5^{\frac{1}{8}}\)[/tex]. Therefore:
[tex]\[ (\sqrt[8]{5})^9 = \left(5^{\frac{1}{8}}\right)^9 = 5^{\frac{1}{8} \cdot 9} = 5^{\frac{9}{8}} \][/tex]
- This is not equivalent to [tex]\( 5^{\frac{3}{2}} \)[/tex].
The expressions that are equivalent to [tex]\( \left(5^{\frac{1}{8}} \cdot 5^{\frac{3}{8}}\right)^3 \)[/tex] are:
- [tex]\( 5^{\frac{3}{2}} \)[/tex]
- [tex]\( \sqrt{5^3} \)[/tex]
