Solve for z 7^(z^2)-49^(6-2z)=0

$7^{z^{2}} - 49^{6 - 2 z} = 0$

Step 1

Move $- 49^{6 - 2 z}$ to the right side of the equation by adding it to both sides.

$7^{z^{2}} = 49^{6 - 2 z}$

Step 2

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

$7^{z^{2}} = 7^{2 (6 - 2 z)}$

Step 3

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

$z^{2} = 2 (6 - 2 z)$

Step 4

Solve for $z$ .

Simplify $2 (6 - 2 z)$ .

Apply the distributive property.

$z^{2} = 2 \cdot 6 + 2 (- 2 z)$

Multiply.

Multiply $2$ by $6$ .

$z^{2} = 12 + 2 (- 2 z)$

Multiply $- 2$ by $2$ .

$z^{2} = 12 - 4 z$

Add $4 z$ to both sides of the equation.

$z^{2} + 4 z = 12$

Subtract $12$ from both sides of the equation.

$z^{2} + 4 z - 12 = 0$

Factor $z^{2} + 4 z - 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $4$ .

$- 2, 6$

Write the factored form using these integers.

$(z - 2) (z + 6) = 0$

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

$z - 2 = 0$

$z + 6 = 0$

Set $z - 2$ equal to $0$ and solve for $z$ .

Set $z - 2$ equal to $0$ .

$z - 2 = 0$

Add $2$ to both sides of the equation.

$z = 2$

Set $z + 6$ equal to $0$ and solve for $z$ .

Set $z + 6$ equal to $0$ .

$z + 6 = 0$

Subtract $6$ from both sides of the equation.

$z = - 6$

The final solution is all the values that make $(z - 2) (z + 6) = 0$ true.

$z = 2, - 6$

Solve for z 7^(z^2)-49^(6-2z)=0

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