sqrt(r^(5))root(6)(r^(7))

Convert to Exponential Form: First, let's convert the radical form to the exponential form. sqrt(r^(5)) = r^(5/2) root(6)(r^(7)) = r^(7/6) sqrt(r^(5))root(6)(r^(7)) = r^(5/2) * r^(7/6)Apply Product Rule of Exponents: Next, we apply the product rule of exponents, which states that a^m * a^n = a^(m+n). Combine the exponents by adding them. r^(5/2) * r^(7/6) = r^(5/2 + 7/6) To add these fractions, we need a common denominator, which is 12. = r^((5*6)/(2*6) + (7*2)/(6*2)) = r^((30/12) + (14/12)) = r^((30 + 14)/12) = r^(44/12)Simplify Exponent: Now, we simplify the exponent by dividing both the numerator and the denominator by their greatest common divisor, which is 4. r^(44/12) = r^(11/3)Convert to Radical Form: Finally, we convert the simplified exponential form back to the radical form. r^(11/3) = root(3)(r^11)

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$\sqrt{r^{5}} \sqrt[6]{r^{7}}$

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Algebra 2

Simplify the product of two radical expressions having same variable

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\sqrt{r^{5}} \sqrt[6]{r^{7}}

Convert to Exponential Form: First, let's convert the radical form to the exponential form. $\newline$ $\sqrt{r^{5}} = r^{\frac{5}{2}}$ $\newline$ $\sqrt[6]{r^{7}} = r^{\frac{7}{6}}$ $\newline$ $\sqrt{r^{5}}\sqrt[6]{r^{7}} = r^{\frac{5}{2}} \times r^{\frac{7}{6}}$
Apply Product Rule of Exponents: Next, we apply the product rule of exponents, which states that $a^m \times a^n = a^{m+n}$ . Combine the exponents by adding them. $r^{5/2} \times r^{7/6} = r^{5/2 + 7/6}$ . To add these fractions, we need a common denominator, which is $12$ . $= r^{(5\times6)/(2\times6) + (7\times2)/(6\times2)} = r^{(30/12) + (14/12)} = r^{(30 + 14)/12} = r^{44/12}$
Simplify Exponent: Now, we simplify the exponent by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ . $\newline$ $r^{44/12}$ $\newline$ = $r^{11/3}$
Convert to Radical Form: Finally, we convert the simplified exponential form back to the radical form. $\newline$ $r^{\frac{11}{3}} = \sqrt[3]{r^{11}}$