To simplify the given mathematical expression [tex]\( 3c \sqrt[2]{a^7} \)[/tex], we can follow these steps:
1. Understand the expression: The expression given is [tex]\( 3c \sqrt{a^7} \)[/tex], where [tex]\( \sqrt{} \)[/tex] denotes the square root.
2. Break down the square root term: The square root of [tex]\( a^7 \)[/tex] can be expressed as:
[tex]\[
\sqrt{a^7} = \sqrt{a^6 \cdot a} = \sqrt{a^6} \cdot \sqrt{a}
\][/tex]
3. Simplify the square root of a power: Since [tex]\( \sqrt{a^6} = a^{6/2} = a^3 \)[/tex], we can rewrite the original expression:
[tex]\[
\sqrt{a^7} = a^3 \sqrt{a}
\][/tex]
4. Substitute this back into the original expression: Replacing [tex]\( \sqrt{a^7} \)[/tex] with [tex]\( a^3 \sqrt{a} \)[/tex] in the expression [tex]\( 3c \sqrt{a^7} \)[/tex], we get:
[tex]\[
3c \sqrt{a^7} = 3c (a^3 \sqrt{a})
\][/tex]
5. Combine the terms: Multiply the constants and like terms together:
[tex]\[
3c (a^3 \sqrt{a}) = 3c a^3 \sqrt{a}
\][/tex]
Thus, the simplified form of the expression [tex]\( 3c \sqrt{a^7} \)[/tex] is:
[tex]\[
3c \sqrt{a^7}
\][/tex]
Therefore, the simplified expression is [tex]\( 3c \sqrt{a^7} \)[/tex]. This shows that there was no further simplification necessary beyond the given answer.
