Solve [(15:5)^{4}]^{3}:[(2^{2}*5*3)*12*2^5:2^4]^10-1^3

$$[ ( 15 : 5 ) ^ { 4 } ] ^ { 3 } : [ ( 2 ^ { 2 } \times 5 \times 3 ) \cdot 12 \cdot 2 ^ { 5 } : 2 ^ { 4 } ] ^ { 10 } - 1 ^ { 3 }$$

Evaluate

$$-\frac{72138957898383359999999999}{72138957898383360000000000}\approx -1$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{\left(\frac{15}{5}\right)^{12}}{\left(\frac{2^{2}\times 5\times 3\times 12\times 2^{5}}{2^{4}}\right)^{10}}-1^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{\left(\frac{15}{5}\right)^{12}}{\left(\frac{2^{7}\times 5\times 3\times 12}{2^{4}}\right)^{10}}-1^{3}$$

Divide 15 by 5 to get 3.

$$\frac{3^{12}}{\left(\frac{2^{7}\times 5\times 3\times 12}{2^{4}}\right)^{10}}-1^{3}$$

Calculate 3 to the power of 12 and get 531441.

$$\frac{531441}{\left(\frac{2^{7}\times 5\times 3\times 12}{2^{4}}\right)^{10}}-1^{3}$$

Cancel out 2^{4} in both numerator and denominator.

$$\frac{531441}{\left(3\times 5\times 12\times 2^{3}\right)^{10}}-1^{3}$$

Multiply 3 and 5 to get 15.

$$\frac{531441}{\left(15\times 12\times 2^{3}\right)^{10}}-1^{3}$$

Multiply 15 and 12 to get 180.

$$\frac{531441}{\left(180\times 2^{3}\right)^{10}}-1^{3}$$

Expand \left(180\times 2^{3}\right)^{10}.

$$\frac{531441}{180^{10}\times \left(2^{3}\right)^{10}}-1^{3}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 10 to get 30.

$$\frac{531441}{180^{10}\times 2^{30}}-1^{3}$$

Calculate 180 to the power of 10 and get 35704672266240000000000.

$$\frac{531441}{35704672266240000000000\times 2^{30}}-1^{3}$$

Calculate 2 to the power of 30 and get 1073741824.

$$\frac{531441}{35704672266240000000000\times 1073741824}-1^{3}$$

Multiply 35704672266240000000000 and 1073741824 to get 38337599924474751221760000000000.

$$\frac{531441}{38337599924474751221760000000000}-1^{3}$$

Reduce the fraction \frac{531441}{38337599924474751221760000000000} to lowest terms by extracting and canceling out 531441.

$$\frac{1}{72138957898383360000000000}-1^{3}$$

Calculate 1 to the power of 3 and get 1.

$$\frac{1}{72138957898383360000000000}-1$$

Subtract 1 from \frac{1}{72138957898383360000000000} to get -\frac{72138957898383359999999999}{72138957898383360000000000}.

$$-\frac{72138957898383359999999999}{72138957898383360000000000}$$

Factor

$$-\frac{72138957898383359999999999}{72138957898383360000000000} = -1$$