Algebra Examples

$(5 a^{2} b^{3}) (6 a^{k} b) = 30 a^{6} b^{4}$

Step 1

Simplify.

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Step 1.1

Multiply $6$ by $5$ .

$30 a^{2} b^{3} (a^{k} b) = 30 a^{6} b^{4}$

Step 1.2

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

$30 a^{k + 2} b^{3} b = 30 a^{6} b^{4}$

Step 1.3

Raise $b$ to the power of $1$ .

$30 a^{k + 2} (b^{1} b^{3}) = 30 a^{6} b^{4}$

Step 1.4

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

$30 a^{k + 2} b^{1 + 3} = 30 a^{6} b^{4}$

Step 1.5

Add $1$ and $3$ .

$30 a^{k + 2} b^{4} = 30 a^{6} b^{4}$

Step 2

Divide each term in $30 a^{k + 2} b^{4} = 30 a^{6} b^{4}$ by $30 b^{4}$ and simplify.

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Step 2.1

Divide each term in $30 a^{k + 2} b^{4} = 30 a^{6} b^{4}$ by $30 b^{4}$ .

$\frac{30 a^{k + 2} b^{4}}{30 b^{4}} = \frac{30 a^{6} b^{4}}{30 b^{4}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $30$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{30 a^{k + 2} b^{4}}{30 b^{4}} = \frac{30 a^{6} b^{4}}{30 b^{4}}$

Step 2.2.1.2

Rewrite the expression.

$\frac{a^{k + 2} b^{4}}{b^{4}} = \frac{30 a^{6} b^{4}}{30 b^{4}}$

Step 2.2.2

Cancel the common factor of $b^{4}$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{a^{k + 2} b^{4}}{b^{4}} = \frac{30 a^{6} b^{4}}{30 b^{4}}$

Step 2.2.2.2

Divide $a^{k + 2}$ by $1$ .

$a^{k + 2} = \frac{30 a^{6} b^{4}}{30 b^{4}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $30$ .

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Step 2.3.1.1

Cancel the common factor.

$a^{k + 2} = \frac{30 a^{6} b^{4}}{30 b^{4}}$

Step 2.3.1.2

Rewrite the expression.

$a^{k + 2} = \frac{a^{6} b^{4}}{b^{4}}$

Step 2.3.2

Cancel the common factor of $b^{4}$ .

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Step 2.3.2.1

Cancel the common factor.

$a^{k + 2} = \frac{a^{6} b^{4}}{b^{4}}$

Step 2.3.2.2

Divide $a^{6}$ by $1$ .

$a^{k + 2} = a^{6}$

Step 3

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

$k + 2 = 6$

Step 4

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

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Step 4.1

Subtract $2$ from both sides of the equation.

$k = 6 - 2$

Step 4.2

Subtract $2$ from $6$ .

$k = 4$