Question
simplify
√f^14g^9h^11
√f^14g^9h^11
Answers
GPT-4o mini
To simplify the expression \(\sqrt{f^{14}g^9h^{11}}\), we can use the property that \(\sqrt{a^b} = a^{b/2}\). Let's apply this property to each term inside the square root:
1. For \(f^{14}\):
\[
\sqrt{f^{14}} = f^{14/2} = f^7
\]
2. For \(g^{9}\):
\[
\sqrt{g^{9}} = g^{9/2} = g^{4.5} = g^4 \cdot g^{1/2}
\]
3. For \(h^{11}\):
\[
\sqrt{h^{11}} = h^{11/2} = h^5 \cdot h^{1/2}
\]
Now, putting it all together:
\[
\sqrt{f^{14}g^9h^{11}} = f^7 \cdot g^4 \cdot g^{1/2} \cdot h^5 \cdot h^{1/2}
\]
This simplifies to:
\[
f^7 g^4 h^5 \sqrt{gh}
\]
Thus, the simplified expression is:
\[
f^7 g^4 h^5 \sqrt{gh}
\]
1. For \(f^{14}\):
\[
\sqrt{f^{14}} = f^{14/2} = f^7
\]
2. For \(g^{9}\):
\[
\sqrt{g^{9}} = g^{9/2} = g^{4.5} = g^4 \cdot g^{1/2}
\]
3. For \(h^{11}\):
\[
\sqrt{h^{11}} = h^{11/2} = h^5 \cdot h^{1/2}
\]
Now, putting it all together:
\[
\sqrt{f^{14}g^9h^{11}} = f^7 \cdot g^4 \cdot g^{1/2} \cdot h^5 \cdot h^{1/2}
\]
This simplifies to:
\[
f^7 g^4 h^5 \sqrt{gh}
\]
Thus, the simplified expression is:
\[
f^7 g^4 h^5 \sqrt{gh}
\]