Answer :
To find which expression is equivalent to [tex]\( g^2 h \sqrt{5 g} \)[/tex], we will simplify the given expression step-by-step.
1. Start with the original expression:
[tex]\[ g^2 h \sqrt{5 g} \][/tex]
2. Notice that the square root can be separated:
[tex]\[ \sqrt{5 g} = \sqrt{5} \cdot \sqrt{g} \][/tex]
3. Now, substitute this back into the original expression:
[tex]\[ g^2 h \cdot \sqrt{5} \cdot \sqrt{g} \][/tex]
4. Recognize that [tex]\(\sqrt{g}\)[/tex] is the same as [tex]\(g^{1/2}\)[/tex]:
[tex]\[ g^2 h \cdot \sqrt{5} \cdot g^{1/2} \][/tex]
5. Combine the powers of [tex]\(g\)[/tex]:
[tex]\[ g^2 \cdot g^{1/2} = g^{2 + 1/2} = g^{5/2} \][/tex]
6. Now the expression becomes:
[tex]\[ g^{5/2} h \cdot \sqrt{5} \][/tex]
7. Rewriting [tex]\(g^{5/2}\)[/tex] in terms of the square root notation:
[tex]\[ g^{5/2} = \sqrt{g^5} \][/tex]
So we can reframe the entire expression as:
[tex]\[ \sqrt{g^5} \cdot h \cdot \sqrt{5} \][/tex]
8. Combine the square roots and [tex]\(h\)[/tex]:
[tex]\[ \sqrt{5 g^5} \cdot h = h \cdot \sqrt{5 g^5} \][/tex]
9. Finally, combine [tex]\(h\)[/tex] inside the square root:
[tex]\[ \sqrt{5 g^5 h} \][/tex]
Hence, the expression [tex]\( g^2 h \sqrt{5 g} \)[/tex] simplifies to [tex]\(\sqrt{5 g^5 h}\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
1. Start with the original expression:
[tex]\[ g^2 h \sqrt{5 g} \][/tex]
2. Notice that the square root can be separated:
[tex]\[ \sqrt{5 g} = \sqrt{5} \cdot \sqrt{g} \][/tex]
3. Now, substitute this back into the original expression:
[tex]\[ g^2 h \cdot \sqrt{5} \cdot \sqrt{g} \][/tex]
4. Recognize that [tex]\(\sqrt{g}\)[/tex] is the same as [tex]\(g^{1/2}\)[/tex]:
[tex]\[ g^2 h \cdot \sqrt{5} \cdot g^{1/2} \][/tex]
5. Combine the powers of [tex]\(g\)[/tex]:
[tex]\[ g^2 \cdot g^{1/2} = g^{2 + 1/2} = g^{5/2} \][/tex]
6. Now the expression becomes:
[tex]\[ g^{5/2} h \cdot \sqrt{5} \][/tex]
7. Rewriting [tex]\(g^{5/2}\)[/tex] in terms of the square root notation:
[tex]\[ g^{5/2} = \sqrt{g^5} \][/tex]
So we can reframe the entire expression as:
[tex]\[ \sqrt{g^5} \cdot h \cdot \sqrt{5} \][/tex]
8. Combine the square roots and [tex]\(h\)[/tex]:
[tex]\[ \sqrt{5 g^5} \cdot h = h \cdot \sqrt{5 g^5} \][/tex]
9. Finally, combine [tex]\(h\)[/tex] inside the square root:
[tex]\[ \sqrt{5 g^5 h} \][/tex]
Hence, the expression [tex]\( g^2 h \sqrt{5 g} \)[/tex] simplifies to [tex]\(\sqrt{5 g^5 h}\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
