Solve n^2=3^7*5^5/15^3

$$n ^ { 2 } = \frac { 3 ^ { 7 } \cdot 5 ^ { 5 } } { 15 ^ { 3 } }$$

Solve for n

$$n=45$$

$$n=-45$$

Steps Using Difference of Squares

Calculate 3 to the power of 7 and get 2187.

$$n^{2}=\frac{2187\times 5^{5}}{15^{3}}$$

Calculate 5 to the power of 5 and get 3125.

$$n^{2}=\frac{2187\times 3125}{15^{3}}$$

Multiply 2187 and 3125 to get 6834375.

$$n^{2}=\frac{6834375}{15^{3}}$$

Calculate 15 to the power of 3 and get 3375.

$$n^{2}=\frac{6834375}{3375}$$

Divide 6834375 by 3375 to get 2025.

$$n^{2}=2025$$

Subtract 2025 from both sides.

$$n^{2}-2025=0$$

Consider n^{2}-2025. Rewrite n^{2}-2025 as n^{2}-45^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(n-45\right)\left(n+45\right)=0$$

To find equation solutions, solve n-45=0 and n+45=0.

$$n=45 n=-45$$

Steps by Finding Square Root

Calculate 3 to the power of 7 and get 2187.

$$n^{2}=\frac{2187\times 5^{5}}{15^{3}}$$

Calculate 5 to the power of 5 and get 3125.

$$n^{2}=\frac{2187\times 3125}{15^{3}}$$

Multiply 2187 and 3125 to get 6834375.

$$n^{2}=\frac{6834375}{15^{3}}$$

Calculate 15 to the power of 3 and get 3375.

$$n^{2}=\frac{6834375}{3375}$$

Divide 6834375 by 3375 to get 2025.

$$n^{2}=2025$$

Take the square root of both sides of the equation.

$$n=45 n=-45$$

Steps Using the Quadratic Formula

Calculate 3 to the power of 7 and get 2187.

$$n^{2}=\frac{2187\times 5^{5}}{15^{3}}$$

Calculate 5 to the power of 5 and get 3125.

$$n^{2}=\frac{2187\times 3125}{15^{3}}$$

Multiply 2187 and 3125 to get 6834375.

$$n^{2}=\frac{6834375}{15^{3}}$$

Calculate 15 to the power of 3 and get 3375.

$$n^{2}=\frac{6834375}{3375}$$

Divide 6834375 by 3375 to get 2025.

$$n^{2}=2025$$

Subtract 2025 from both sides.

$$n^{2}-2025=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -2025 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-2025\right)}}{2}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-2025\right)}}{2}$$

Multiply -4 times -2025.

$$n=\frac{0±\sqrt{8100}}{2}$$

Take the square root of 8100.

$$n=\frac{0±90}{2}$$

Now solve the equation n=\frac{0±90}{2} when ± is plus. Divide 90 by 2.

$$n=45$$

Now solve the equation n=\frac{0±90}{2} when ± is minus. Divide -90 by 2.

$$n=-45$$

The equation is now solved.

$$n=45 n=-45$$