Algebra Examples

$\frac{p^{10 x^{2}}}{p^{5 x}} = {(p^{- 20})}^{x}$

Step 1

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

$\frac{p^{10 x^{2}}}{p^{5 x}} = p^{- 20 x}$

Step 2

Move $p^{5 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$p^{10 x^{2}} p^{- 5 x} = p^{- 20 x}$

Step 3

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

$p^{10 x^{2} - 5 x} = p^{- 20 x}$

Step 4

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

$10 x^{2} - 5 x = - 20 x$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Add $20 x$ to both sides of the equation.

$10 x^{2} - 5 x + 20 x = 0$

Step 5.1.2

Add $- 5 x$ and $20 x$ .

$10 x^{2} + 15 x = 0$

Step 5.2

Factor $5 x$ out of $10 x^{2} + 15 x$ .

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Step 5.2.1

Factor $5 x$ out of $10 x^{2}$ .

$5 x (2 x) + 15 x = 0$

Step 5.2.2

Factor $5 x$ out of $15 x$ .

$5 x (2 x) + 5 x (3) = 0$

Step 5.2.3

Factor $5 x$ out of $5 x (2 x) + 5 x (3)$ .

$5 x (2 x + 3) = 0$

Step 5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 x + 3 = 0$

Step 5.4

Set $x$ equal to $0$ .

$x = 0$

Step 5.5

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 5.5.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 5.5.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 5.5.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 5.5.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.5.2.2.2

Simplify the left side.

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Step 5.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 5.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

Step 5.5.2.2.3

Simplify the right side.

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Step 5.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 5.6

The final solution is all the values that make $5 x (2 x + 3) = 0$ true.

$x = 0, - \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{3}{2}$

Decimal Form:

$x = 0, - 1.5$

Mixed Number Form:

$x = 0, - 1 \frac{1}{2}$