Solve [(1-a)(1+a+a^2)(1+a^3)-1]:(-a)^5

$$[ ( 1 - a ) ( 1 + a + a ^ { 2 } ) ( 1 + a ^ { 3 } ) - 1 ] : ( - a ) ^ { 5 }$$

Evaluate

$$a$$

Short Solution Steps

Use the distributive property to multiply 1-a by 1+a+a^{2} and combine like terms.

$$\frac{\left(1-a^{3}\right)\left(1+a^{3}\right)-1}{\left(-a\right)^{5}}$$

Consider \left(1-a^{3}\right)\left(1+a^{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\frac{1-\left(a^{3}\right)^{2}-1}{\left(-a\right)^{5}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{1-a^{6}-1}{\left(-a\right)^{5}}$$

Subtract 1 from 1 to get 0.

$$\frac{-a^{6}}{\left(-a\right)^{5}}$$

Expand \left(-a\right)^{5}.

$$\frac{-a^{6}}{\left(-1\right)^{5}a^{5}}$$

Calculate -1 to the power of 5 and get -1.

$$\frac{-a^{6}}{-a^{5}}$$

Cancel out -a^{5} in both numerator and denominator.

$$a$$

Differentiate w.r.t. a

$$1$$

Steps Using Definition of a Derivative

Use the distributive property to multiply 1-a by 1+a+a^{2} and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\left(1-a^{3}\right)\left(1+a^{3}\right)-1}{\left(-a\right)^{5}})$$

Consider \left(1-a^{3}\right)\left(1+a^{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1-\left(a^{3}\right)^{2}-1}{\left(-a\right)^{5}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1-a^{6}-1}{\left(-a\right)^{5}})$$

Subtract 1 from 1 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{-a^{6}}{\left(-a\right)^{5}})$$

Expand \left(-a\right)^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{-a^{6}}{\left(-1\right)^{5}a^{5}})$$

Calculate -1 to the power of 5 and get -1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{-a^{6}}{-a^{5}})$$

Cancel out -a^{5} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a)$$

The derivative of ax^{n} is nax^{n-1}.

$$a^{1-1}$$

Subtract 1 from 1.

$$a^{0}$$

For any term t except 0, t^{0}=1.

$$1$$