Solve frac{n^{5}*p^{-3}}{(n/p)^-2}div(p^2n^4)^+1

$$\frac { n ^ { 5 } \cdot p ^ { - 3 } } { ( n / p ) ^ { - 2 } } \div ( p ^ { 2 } n ^ { 4 } ) ^ { + 1 }$$

Evaluate

$$\frac{n^{3}}{p^{7}}$$

Solution Steps

To raise \frac{n}{p} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{n^{5}p^{-3}}{\frac{n^{-2}}{p^{-2}}}}{\left(p^{2}n^{4}\right)^{1}}$$

Divide n^{5}p^{-3} by \frac{n^{-2}}{p^{-2}} by multiplying n^{5}p^{-3} by the reciprocal of \frac{n^{-2}}{p^{-2}}.

$$\frac{\frac{n^{5}p^{-3}p^{-2}}{n^{-2}}}{\left(p^{2}n^{4}\right)^{1}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{p^{-3}p^{-2}n^{7}}{\left(p^{2}n^{4}\right)^{1}}$$

To multiply powers of the same base, add their exponents. Add -3 and -2 to get -5.

$$\frac{p^{-5}n^{7}}{\left(p^{2}n^{4}\right)^{1}}$$

Calculate p^{2}n^{4} to the power of 1 and get p^{2}n^{4}.

$$\frac{p^{-5}n^{7}}{p^{2}n^{4}}$$

Cancel out n^{4} in both numerator and denominator.

$$\frac{p^{-5}n^{3}}{p^{2}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{n^{3}}{p^{7}}$$

Expand

$$\frac{n^{3}}{p^{7}}$$

Solution Steps

To raise \frac{n}{p} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{n^{5}p^{-3}}{\frac{n^{-2}}{p^{-2}}}}{\left(p^{2}n^{4}\right)^{1}}$$

Divide n^{5}p^{-3} by \frac{n^{-2}}{p^{-2}} by multiplying n^{5}p^{-3} by the reciprocal of \frac{n^{-2}}{p^{-2}}.

$$\frac{\frac{n^{5}p^{-3}p^{-2}}{n^{-2}}}{\left(p^{2}n^{4}\right)^{1}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{p^{-3}p^{-2}n^{7}}{\left(p^{2}n^{4}\right)^{1}}$$

To multiply powers of the same base, add their exponents. Add -3 and -2 to get -5.

$$\frac{p^{-5}n^{7}}{\left(p^{2}n^{4}\right)^{1}}$$

Calculate p^{2}n^{4} to the power of 1 and get p^{2}n^{4}.

$$\frac{p^{-5}n^{7}}{p^{2}n^{4}}$$

Cancel out n^{4} in both numerator and denominator.

$$\frac{p^{-5}n^{3}}{p^{2}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{n^{3}}{p^{7}}$$