Algebra Examples

$M = \frac{10^{m + n} \cdot 10^{n - m} \cdot 10^{n + 1}}{10^{n - 1} \cdot 10^{2 n + 1}}$

Step 1

Multiply $10^{m + n}$ by $10^{n - m}$ by adding the exponents.

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Step 1.1

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

$M = \frac{10^{m + n + n - m} \cdot 10^{n + 1}}{10^{n - 1} \cdot 10^{2 n + 1}}$

Step 1.2

Combine the opposite terms in $m + n + n - m$ .

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Step 1.2.1

Subtract $m$ from $m$ .

$M = \frac{10^{n + n + 0} \cdot 10^{n + 1}}{10^{n - 1} \cdot 10^{2 n + 1}}$

Step 1.2.2

Add $n + n$ and $0$ .

$M = \frac{10^{n + n} \cdot 10^{n + 1}}{10^{n - 1} \cdot 10^{2 n + 1}}$

Step 1.3

Add $n$ and $n$ .

$M = \frac{10^{2 n} \cdot 10^{n + 1}}{10^{n - 1} \cdot 10^{2 n + 1}}$

Step 2

Multiply $10^{2 n}$ by $10^{n + 1}$ by adding the exponents.

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Step 2.1

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

$M = \frac{10^{2 n + n + 1}}{10^{n - 1} \cdot 10^{2 n + 1}}$

Step 2.2

Add $2 n$ and $n$ .

$M = \frac{10^{3 n + 1}}{10^{n - 1} \cdot 10^{2 n + 1}}$

Step 3

Multiply $10^{n - 1}$ by $10^{2 n + 1}$ by adding the exponents.

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Step 3.1

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

$M = \frac{10^{3 n + 1}}{10^{n - 1 + 2 n + 1}}$

Step 3.2

Combine the opposite terms in $n - 1 + 2 n + 1$ .

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Step 3.2.1

Add $- 1$ and $1$ .

$M = \frac{10^{3 n + 1}}{10^{n + 2 n + 0}}$

Step 3.2.2

Add $n + 2 n$ and $0$ .

$M = \frac{10^{3 n + 1}}{10^{n + 2 n}}$

Step 3.3

Add $n$ and $2 n$ .

$M = \frac{10^{3 n + 1}}{10^{3 n}}$

Step 4

Cancel the common factor of $10^{3 n + 1}$ and $10^{3 n}$ .

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Step 4.1

Factor $10^{3 n}$ out of $10^{3 n + 1}$ .

$M = \frac{10^{3 n} \cdot 10}{10^{3 n}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$M = \frac{10^{3 n} \cdot 10}{10^{3 n} \cdot 1}$

Step 4.2.2

Cancel the common factor.

$M = \frac{10^{3 n} \cdot 10}{10^{3 n} \cdot 1}$

Step 4.2.3

Rewrite the expression.

$M = \frac{10}{1}$

Step 4.2.4

Divide $10$ by $1$ .

$M = 10$