Simplify (m*m^-2n^(5/3))^2

${(m \cdot m^{- 2} n^{\frac{5}{3}})}^{2}$

Step 1

Multiply $m$ by $m^{- 2}$ by adding the exponents.

Move $m^{- 2}$ .

${(m^{- 2} m \cdot n^{\frac{5}{3}})}^{2}$

Multiply $m^{- 2}$ by $m$ .

Raise $m$ to the power of $1$ .

${(m^{- 2} m^{1} \cdot n^{\frac{5}{3}})}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(m^{- 2 + 1} \cdot n^{\frac{5}{3}})}^{2}$

Add $- 2$ and $1$ .

${(m^{- 1} \cdot n^{\frac{5}{3}})}^{2}$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{m} \cdot n^{\frac{5}{3}})}^{2}$

Step 3

Combine $\frac{1}{m}$ and $n^{\frac{5}{3}}$ .

${(\frac{n^{\frac{5}{3}}}{m})}^{2}$

Step 4

Apply basic rules of exponents.

Apply the product rule to $\frac{n^{\frac{5}{3}}}{m}$ .

$\frac{{(n^{\frac{5}{3}})}^{2}}{m^{2}}$

Multiply the exponents in ${(n^{\frac{5}{3}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{n^{\frac{5}{3} \cdot 2}}{m^{2}}$

Multiply $\frac{5}{3} \cdot 2$ .

Combine $\frac{5}{3}$ and $2$ .

$\frac{n^{\frac{5 \cdot 2}{3}}}{m^{2}}$

Multiply $5$ by $2$ .

$\frac{n^{\frac{10}{3}}}{m^{2}}$