Algebra Examples

$3^{x} \cdot 81^{5001} = 9^{2 x} \cdot 27^{2 x + 602}$

Step 1

Rewrite $81$ as $3^{4}$ .

$3^{x} \cdot {(3^{4})}^{5001} = 9^{2 x} \cdot 27^{2 x + 602}$

Step 2

Multiply the exponents in ${(3^{4})}^{5001}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{x} \cdot 3^{4 \cdot 5001} = 9^{2 x} \cdot 27^{2 x + 602}$

Step 2.2

Multiply $4$ by $5001$ .

$3^{x} \cdot 3^{20004} = 9^{2 x} \cdot 27^{2 x + 602}$

Step 3

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

$3^{x + 20004} = 9^{2 x} \cdot 27^{2 x + 602}$

Step 4

Rewrite $9$ as $3^{2}$ .

$3^{x + 20004} = {(3^{2})}^{2 x} \cdot 27^{2 x + 602}$

Step 5

Multiply the exponents in ${(3^{2})}^{2 x}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{x + 20004} = 3^{2 (2 x)} \cdot 27^{2 x + 602}$

Step 5.2

Multiply $2$ by $2$ .

$3^{x + 20004} = 3^{4 x} \cdot 27^{2 x + 602}$

Step 6

Rewrite $27$ as $3^{3}$ .

$3^{x + 20004} = 3^{4 x} \cdot {(3^{3})}^{2 x + 602}$

Step 7

Multiply the exponents in ${(3^{3})}^{2 x + 602}$ .

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Step 7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{x + 20004} = 3^{4 x} \cdot 3^{3 (2 x + 602)}$

Step 7.2

Apply the distributive property.

$3^{x + 20004} = 3^{4 x} \cdot 3^{3 (2 x) + 3 \cdot 602}$

Step 7.3

Multiply $2$ by $3$ .

$3^{x + 20004} = 3^{4 x} \cdot 3^{6 x + 3 \cdot 602}$

Step 7.4

Multiply $3$ by $602$ .

$3^{x + 20004} = 3^{4 x} \cdot 3^{6 x + 1806}$

Step 8

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

$3^{x + 20004} = 3^{4 x + 6 x + 1806}$

Step 9

Add $4 x$ and $6 x$ .

$3^{x + 20004} = 3^{10 x + 1806}$

Step 10

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x + 20004 = 10 x + 1806$

Step 11

Solve for $x$ .

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Step 11.1

Move all terms containing $x$ to the left side of the equation.

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Step 11.1.1

Subtract $10 x$ from both sides of the equation.

$x + 20004 - 10 x = 1806$

Step 11.1.2

Subtract $10 x$ from $x$ .

$- 9 x + 20004 = 1806$

Step 11.2

Move all terms not containing $x$ to the right side of the equation.

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Step 11.2.1

Subtract $20004$ from both sides of the equation.

$- 9 x = 1806 - 20004$

Step 11.2.2

Subtract $20004$ from $1806$ .

$- 9 x = - 18198$

Step 11.3

Divide each term in $- 9 x = - 18198$ by $- 9$ and simplify.

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Step 11.3.1

Divide each term in $- 9 x = - 18198$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{- 18198}{- 9}$

Step 11.3.2

Simplify the left side.

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Step 11.3.2.1

Cancel the common factor of $- 9$ .

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Step 11.3.2.1.1

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{- 18198}{- 9}$

Step 11.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 18198}{- 9}$

Step 11.3.3

Simplify the right side.

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Step 11.3.3.1

Divide $- 18198$ by $- 9$ .

$x = 2022$