Question

what is the simplest form of $sqrt{a^{7}}$?
a. $a^{2}sqrt{a}$
b. $a^{3}sqrt{a}$
c. $a^{3}sqrt{a^{2}}$
d. $3asqrt{a}$

Explanation:

Step1: Rewrite the exponent

We know that $\sqrt{a^{7}}=(a^{7})^{\frac{1}{2}}$. By the power - of - a - power rule $(x^{m})^{n}=x^{mn}$, so $(a^{7})^{\frac{1}{2}}=a^{7\times\frac{1}{2}} = a^{\frac{7}{2}}$.

Step2: Split the exponent

We can write $a^{\frac{7}{2}}$ as $a^{3+\frac{1}{2}}$. By the rule $x^{m + n}=x^{m}\times x^{n}$, we have $a^{3+\frac{1}{2}}=a^{3}\times a^{\frac{1}{2}}$. Since $a^{\frac{1}{2}}=\sqrt{a}$, then $a^{3}\times a^{\frac{1}{2}}=a^{3}\sqrt{a}$.

Answer:

B. $a^{3}\sqrt{a}$