Solve for P 3x^P(4x^(P+4)+2x^(3p+2))=12x^12+6x^18

$3 x^{P} (4 x^{P + 4} + 2 x^{3 p + 2}) = 12 x^{12} + 6 x^{18}$

Step 1

Simplify $3 x^{P} (4 x^{P + 4} + 2 x^{3 p + 2})$ .

Simplify by multiplying through.

Apply the distributive property.

$3 x^{P} (4 x^{P + 4}) + 3 x^{P} (2 x^{3 p + 2}) = 12 x^{12} + 6 x^{18}$

Reorder.

Rewrite using the commutative property of multiplication.

$3 \cdot 4 x^{P} x^{P + 4} + 3 x^{P} (2 x^{3 p + 2}) = 12 x^{12} + 6 x^{18}$

Rewrite using the commutative property of multiplication.

$3 \cdot 4 x^{P} x^{P + 4} + 3 \cdot 2 x^{P} x^{3 p + 2} = 12 x^{12} + 6 x^{18}$

Simplify each term.

Multiply $x^{P}$ by $x^{P + 4}$ by adding the exponents.

Move $x^{P + 4}$ .

$3 \cdot 4 (x^{P + 4} x^{P}) + 3 \cdot 2 x^{P} x^{3 p + 2} = 12 x^{12} + 6 x^{18}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 \cdot 4 x^{P + 4 + P} + 3 \cdot 2 x^{P} x^{3 p + 2} = 12 x^{12} + 6 x^{18}$

Add $P$ and $P$ .

$3 \cdot 4 x^{2 P + 4} + 3 \cdot 2 x^{P} x^{3 p + 2} = 12 x^{12} + 6 x^{18}$

Multiply $3$ by $4$ .

$12 x^{2 P + 4} + 3 \cdot 2 x^{P} x^{3 p + 2} = 12 x^{12} + 6 x^{18}$

Multiply $x^{P}$ by $x^{3 p + 2}$ by adding the exponents.

Move $x^{3 p + 2}$ .

$12 x^{2 P + 4} + 3 \cdot 2 (x^{3 p + 2} x^{P}) = 12 x^{12} + 6 x^{18}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{2 P + 4} + 3 \cdot 2 x^{3 p + 2 + P} = 12 x^{12} + 6 x^{18}$

Multiply $3$ by $2$ .

$12 x^{2 P + 4} + 6 x^{3 p + 2 + P} = 12 x^{12} + 6 x^{18}$

Step 2

Rewrite $x^{2 P + 4}$ as $x^{2 P} x^{4}$ .

$12 (x^{2 P} x^{4}) + 6 x^{3 p + 2 + P} = 12 x^{12} + 6 x^{18}$

Step 3

Rewrite $x^{3 p + 2 + P}$ as $x^{3 p + 2} x^{P}$ .

$12 (x^{2 P} x^{4}) + 6 (x^{3 p + 2} x^{P}) = 12 x^{12} + 6 x^{18}$

Step 4

Rewrite $x^{3 p + 2}$ as $x^{3 p} x^{2}$ .

$12 (x^{2 P} x^{4}) + 6 (x^{3 p} x^{2} x^{P}) = 12 x^{12} + 6 x^{18}$

Step 5

Rewrite $x^{2 P}$ as exponentiation.

$12 ({(x^{P})}^{2} x^{4}) + 6 (x^{3 p} x^{2} x^{P}) = 12 x^{12} + 6 x^{18}$

Step 6

Rewrite $x^{3 p}$ as exponentiation.

$12 ({(x^{P})}^{2} x^{4}) + 6 ({(x^{p})}^{3} x^{2} x^{P}) = 12 x^{12} + 6 x^{18}$

Step 7

Remove parentheses.

$12 {(x^{P})}^{2} x^{4} + 6 {(x^{p})}^{3} x^{2} x^{P} = 12 x^{12} + 6 x^{18}$

Solve for P 3x^P(4x^(P+4)+2x^(3p+2))=12x^12+6x^18

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