Basic Math Examples

$\frac{r^{3}}{r^{8}} = \frac{r^{2}}{r^{7}}$

Step 1

Factor each term.

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Step 1.1

Reduce the expression $\frac{r^{3}}{r^{8}}$ by cancelling the common factors.

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Step 1.1.1

Multiply by $1$ .

$\frac{r^{3} \cdot 1}{r^{8}} = \frac{r^{2}}{r^{7}}$

Step 1.1.2

Factor $r^{3}$ out of $r^{8}$ .

$\frac{r^{3} \cdot 1}{r^{3} r^{5}} = \frac{r^{2}}{r^{7}}$

Step 1.1.3

Cancel the common factor.

$\frac{r^{3} \cdot 1}{r^{3} r^{5}} = \frac{r^{2}}{r^{7}}$

Step 1.1.4

Rewrite the expression.

$\frac{1}{r^{5}} = \frac{r^{2}}{r^{7}}$

Step 1.2

Reduce the expression $\frac{r^{2}}{r^{7}}$ by cancelling the common factors.

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Step 1.2.1

Multiply by $1$ .

$\frac{1}{r^{5}} = \frac{r^{2} \cdot 1}{r^{7}}$

Step 1.2.2

Factor $r^{2}$ out of $r^{7}$ .

$\frac{1}{r^{5}} = \frac{r^{2} \cdot 1}{r^{2} r^{5}}$

Step 1.2.3

Cancel the common factor.

$\frac{1}{r^{5}} = \frac{r^{2} \cdot 1}{r^{2} r^{5}}$

Step 1.2.4

Rewrite the expression.

$\frac{1}{r^{5}} = \frac{1}{r^{5}}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$r^{5}, r^{5}$

Step 2.2

Since $r^{5}, r^{5}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1$ then find LCM for the variable part $r^{5}, r^{5}$ .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.6

The factors for $r^{5}$ are $r \cdot r \cdot r \cdot r \cdot r$ , which is $r$ multiplied by each other $5$ times.

$r^{5} = r \cdot r \cdot r \cdot r \cdot r$

$r$ occurs $5$ times.

Step 2.7

The LCM of $r^{5}, r^{5}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$r \cdot r \cdot r \cdot r \cdot r$

Step 2.8

Simplify $r \cdot r \cdot r \cdot r \cdot r$ .

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Step 2.8.1

Multiply $r$ by $r$ .

$r^{2} \cdot r \cdot r \cdot r$

Step 2.8.2

Multiply $r^{2}$ by $r$ by adding the exponents.

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Step 2.8.2.1

Multiply $r^{2}$ by $r$ .

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Step 2.8.2.1.1

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

$r^{2} \cdot r^{1} \cdot r \cdot r$

Step 2.8.2.1.2

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

$r^{2 + 1} \cdot r \cdot r$

Step 2.8.2.2

Add $2$ and $1$ .

$r^{3} \cdot r \cdot r$

Step 2.8.3

Multiply $r^{3}$ by $r$ by adding the exponents.

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Step 2.8.3.1

Multiply $r^{3}$ by $r$ .

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Step 2.8.3.1.1

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

$r^{3} \cdot r^{1} \cdot r$

Step 2.8.3.1.2

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

$r^{3 + 1} \cdot r$

Step 2.8.3.2

Add $3$ and $1$ .

$r^{4} \cdot r$

Step 2.8.4

Multiply $r^{4}$ by $r$ by adding the exponents.

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Step 2.8.4.1

Multiply $r^{4}$ by $r$ .

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Step 2.8.4.1.1

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

$r^{4} \cdot r^{1}$

Step 2.8.4.1.2

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

$r^{4 + 1}$

Step 2.8.4.2

Add $4$ and $1$ .

$r^{5}$

Step 3

Multiply each term in $\frac{1}{r^{5}} = \frac{1}{r^{5}}$ by $r^{5}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{r^{5}} = \frac{1}{r^{5}}$ by $r^{5}$ .

$\frac{1}{r^{5}} r^{5} = \frac{1}{r^{5}} r^{5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $r^{5}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1}{r^{5}} r^{5} = \frac{1}{r^{5}} r^{5}$

Step 3.2.1.2

Rewrite the expression.

$1 = \frac{1}{r^{5}} r^{5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $r^{5}$ .

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Step 3.3.1.1

Cancel the common factor.

$1 = \frac{1}{r^{5}} r^{5}$

Step 3.3.1.2

Rewrite the expression.

$1 = 1$

Step 4

Since $1 = 1$ , the equation will always be true for any value of $r$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$