Precalculus Examples

${(\frac{1}{5000})}^{- 2 z} \cdot 5000^{- 2 z + 2} = 5000$

Step 1

Apply the product rule to $\frac{1}{5000}$ .

$\frac{1^{- 2 z}}{5000^{- 2 z}} \cdot 5000^{- 2 z + 2} = 5000$

Step 2

One to any power is one.

$\frac{1}{5000^{- 2 z}} \cdot 5000^{- 2 z + 2} = 5000$

Step 3

Move $5000^{- 2 z}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$5000^{2 z} \cdot 5000^{- 2 z + 2} = 5000$

Step 4

Multiply $5000^{2 z}$ by $5000^{- 2 z + 2}$ by adding the exponents.

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Step 4.1

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

$5000^{2 z - 2 z + 2} = 5000$

Step 4.2

Subtract $2 z$ from $2 z$ .

$5000^{0 + 2} = 5000$

Step 4.3

Add $0$ and $2$ .

$5000^{2} = 5000$

Step 5

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

$5000^{2} = 5000^{1}$

Step 6

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

$2 = 1$

Step 7

Since $2 \neq 1$ , there are no solutions.

No solution