Simplify (a+bx^3)^2dx

${(a + b x^{3})}^{2} d x$

Step 1

Rewrite

${(a + b x^{3})}^{2}$ as

$(a + b x^{3}) (a + b x^{3})$ .

$(a + b x^{3}) (a + b x^{3}) d x$

Step 2

Expand

$(a + b x^{3}) (a + b x^{3})$ using the FOIL Method.

Apply the distributive property.

$(a (a + b x^{3}) + b x^{3} (a + b x^{3})) d x$

Apply the distributive property.

$(a \cdot a + a (b x^{3}) + b x^{3} (a + b x^{3})) d x$

Apply the distributive property.

$(a \cdot a + a (b x^{3}) + b x^{3} a + b x^{3} (b x^{3})) d x$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply

$a$ by

$a$ .

$(a^{2} + a (b x^{3}) + b x^{3} a + b x^{3} (b x^{3})) d x$

Multiply

$b$ by

$b$ by adding the exponents.

Move

$b$ .

$(a^{2} + a b x^{3} + b x^{3} a + b \cdot b x^{3} x^{3}) d x$

Multiply

$b$ by

$b$ .

$(a^{2} + a b x^{3} + b x^{3} a + b^{2} x^{3} x^{3}) d x$

Multiply

$x^{3}$ by

$x^{3}$ by adding the exponents.

Move

$x^{3}$ .

$(a^{2} + a b x^{3} + b x^{3} a + b^{2} (x^{3} x^{3})) d x$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(a^{2} + a b x^{3} + b x^{3} a + b^{2} x^{3 + 3}) d x$

Add

$3$ and

$3$ .

$(a^{2} + a b x^{3} + b x^{3} a + b^{2} x^{6}) d x$

Add

$a b x^{3}$ and

$b x^{3} a$ .

Reorder

$b$ and

$a$ .

$(a^{2} + a b x^{3} + a b x^{3} + b^{2} x^{6}) d x$

Add

$a b x^{3}$ and

$a b x^{3}$ .

$(a^{2} + 2 a b x^{3} + b^{2} x^{6}) d x$

Step 4

Apply the distributive property.

$(a^{2} d + 2 a b x^{3} d + b^{2} x^{6} d) x$

Step 5

Apply the distributive property.

$a^{2} d x + 2 a b x^{3} d x + b^{2} x^{6} d x$

Step 6

Simplify.

Multiply

$x^{3}$ by

$x$ by adding the exponents.

Move

$x$ .

$a^{2} d x + 2 a b (x \cdot x^{3}) d + b^{2} x^{6} d x$

Multiply

$x$ by

$x^{3}$ .

Raise

$x$ to the power of

$1$ .

$a^{2} d x + 2 a b (x^{1} x^{3}) d + b^{2} x^{6} d x$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{2} d x + 2 a b x^{1 + 3} d + b^{2} x^{6} d x$

Add

$1$ and

$3$ .

$a^{2} d x + 2 a b x^{4} d + b^{2} x^{6} d x$

Multiply

$x^{6}$ by

$x$ by adding the exponents.

Move

$x$ .

$a^{2} d x + 2 a b x^{4} d + b^{2} (x \cdot x^{6}) d$

Multiply

$x$ by

$x^{6}$ .

Raise

$x$ to the power of

$1$ .

$a^{2} d x + 2 a b x^{4} d + b^{2} (x^{1} x^{6}) d$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{2} d x + 2 a b x^{4} d + b^{2} x^{1 + 6} d$

Add

$1$ and

$6$ .

$a^{2} d x + 2 a b x^{4} d + b^{2} x^{7} d$

Simplify (a+bx^3)^2dx

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