Solve a(3+sqrt{2})(3-sqrt{2})=

$$a ( 3 + \sqrt { 2 } ) ( 3 - \sqrt { 2 } ) =$$

Evaluate

$$7a$$

Solution Steps

Use the distributive property to multiply a by 3+\sqrt{2}.

$$\left(3a+a\sqrt{2}\right)\left(3-\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of 3a+a\sqrt{2} by each term of 3-\sqrt{2}.

$$9a-3\sqrt{2}a+3a\sqrt{2}-a\left(\sqrt{2}\right)^{2}$$

Combine -3\sqrt{2}a and 3a\sqrt{2} to get 0.

$$9a-a\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$9a-a\times 2$$

Multiply -1 and 2 to get -2.

$$9a-2a$$

Combine 9a and -2a to get 7a.

$$7a$$

Differentiate w.r.t. a

$$7$$

Steps Using Definition of a Derivative

Use the distributive property to multiply a by 3+\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(3a+a\sqrt{2}\right)\left(3-\sqrt{2}\right))$$

Apply the distributive property by multiplying each term of 3a+a\sqrt{2} by each term of 3-\sqrt{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(9a-3\sqrt{2}a+3a\sqrt{2}-a\left(\sqrt{2}\right)^{2})$$

Combine -3\sqrt{2}a and 3a\sqrt{2} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}a}(9a-a\left(\sqrt{2}\right)^{2})$$

The square of \sqrt{2} is 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(9a-a\times 2)$$

Multiply -1 and 2 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(9a-2a)$$

Combine 9a and -2a to get 7a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(7a)$$

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

$$7a^{1-1}$$

Subtract 1 from 1.

$$7a^{0}$$

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

$$7\times 1$$

For any term t, t\times 1=t and 1t=t.

$$7$$