Trigonometry Examples

$8^{1 - p} = 16^{2 p - 1}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$2^{3 (1 - p)} = 2^{4 (2 p - 1)}$

Step 2

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

$3 (1 - p) = 4 (2 p - 1)$

Step 3

Solve for $p$ .

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Step 3.1

Simplify $3 (1 - p)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 3 (1 - p) = 4 (2 p - 1)$

Step 3.1.2

Simplify by adding zeros.

$3 (1 - p) = 4 (2 p - 1)$

Step 3.1.3

Apply the distributive property.

$3 \cdot 1 + 3 (- p) = 4 (2 p - 1)$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $3$ by $1$ .

$3 + 3 (- p) = 4 (2 p - 1)$

Step 3.1.4.2

Multiply $- 1$ by $3$ .

$3 - 3 p = 4 (2 p - 1)$

Step 3.2

Simplify $4 (2 p - 1)$ .

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Step 3.2.1

Apply the distributive property.

$3 - 3 p = 4 (2 p) + 4 \cdot - 1$

Step 3.2.2

Multiply.

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Step 3.2.2.1

Multiply $2$ by $4$ .

$3 - 3 p = 8 p + 4 \cdot - 1$

Step 3.2.2.2

Multiply $4$ by $- 1$ .

$3 - 3 p = 8 p - 4$

Step 3.3

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

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Step 3.3.1

Subtract $8 p$ from both sides of the equation.

$3 - 3 p - 8 p = - 4$

Step 3.3.2

Subtract $8 p$ from $- 3 p$ .

$3 - 11 p = - 4$

Step 3.4

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

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Step 3.4.1

Subtract $3$ from both sides of the equation.

$- 11 p = - 4 - 3$

Step 3.4.2

Subtract $3$ from $- 4$ .

$- 11 p = - 7$

Step 3.5

Divide each term in $- 11 p = - 7$ by $- 11$ and simplify.

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Step 3.5.1

Divide each term in $- 11 p = - 7$ by $- 11$ .

$\frac{- 11 p}{- 11} = \frac{- 7}{- 11}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $- 11$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{- 11 p}{- 11} = \frac{- 7}{- 11}$

Step 3.5.2.1.2

Divide $p$ by $1$ .

$p = \frac{- 7}{- 11}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Dividing two negative values results in a positive value.

$p = \frac{7}{11}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$p = \frac{7}{11}$

Decimal Form:

$p = 0.\overline{63}$