Algebra Examples

$(r^{- 5 x^{2} - 28}) (r^{9}) = \frac{r^{4 x^{2}}}{r^{100}}$

Step 1

Move $r^{100}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$r^{- 5 x^{2} - 28} (r^{9}) = r^{4 x^{2}} r^{- 100}$

Step 2

Multiply $r^{- 5 x^{2} - 28}$ by $r^{9}$ by adding the exponents.

Tap for more steps...

Step 2.1

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

$r^{- 5 x^{2} - 28 + 9} = r^{4 x^{2}} r^{- 100}$

Step 2.2

Add $- 28$ and $9$ .

$r^{- 5 x^{2} - 19} = r^{4 x^{2}} r^{- 100}$

Step 3

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

$r^{- 5 x^{2} - 19} = r^{4 x^{2} - 100}$

Step 4

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

$- 5 x^{2} - 19 = 4 x^{2} - 100$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Subtract $4 x^{2}$ from both sides of the equation.

$- 5 x^{2} - 19 - 4 x^{2} = - 100$

Step 5.1.2

Subtract $4 x^{2}$ from $- 5 x^{2}$ .

$- 9 x^{2} - 19 = - 100$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Add $19$ to both sides of the equation.

$- 9 x^{2} = - 100 + 19$

Step 5.2.2

Add $- 100$ and $19$ .

$- 9 x^{2} = - 81$

Step 5.3

Divide each term in $- 9 x^{2} = - 81$ by $- 9$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $- 9 x^{2} = - 81$ by $- 9$ .

$\frac{- 9 x^{2}}{- 9} = \frac{- 81}{- 9}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $- 9$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 9 x^{2}}{- 9} = \frac{- 81}{- 9}$

Step 5.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 81}{- 9}$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Divide $- 81$ by $- 9$ .

$x^{2} = 9$

Step 5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 5.5

Simplify $\pm \sqrt{9}$ .

Tap for more steps...

Step 5.5.1

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

$x = \pm \sqrt{3^{2}}$

Step 5.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 5.6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 5.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 5.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 5.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$