Find the Domain 6(y+7)gggggggg=3y

$6 (y + 7) g \cdot g \cdot g \cdot g \cdot g \cdot g \cdot g \cdot g = 3 y$

Step 1

Simplify $6 (y + 7) g \cdot g \cdot g \cdot g \cdot g \cdot g \cdot g \cdot g$ .

Multiply $g$ by $g$ by adding the exponents.

Move $g$ .

$6 (y + 7) (g \cdot g) g \cdot g \cdot g \cdot g \cdot g \cdot g = 3 y$

Multiply $g$ by $g$ .

$6 (y + 7) g^{2} g \cdot g \cdot g \cdot g \cdot g \cdot g = 3 y$

Multiply $g^{2}$ by $g$ by adding the exponents.

Move $g$ .

$6 (y + 7) (g \cdot g^{2}) g \cdot g \cdot g \cdot g \cdot g = 3 y$

Multiply $g$ by $g^{2}$ .

Raise $g$ to the power of $1$ .

$6 (y + 7) (g^{1} g^{2}) g \cdot g \cdot g \cdot g \cdot g = 3 y$

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

$6 (y + 7) g^{1 + 2} g \cdot g \cdot g \cdot g \cdot g = 3 y$

Add $1$ and $2$ .

$6 (y + 7) g^{3} g \cdot g \cdot g \cdot g \cdot g = 3 y$

Multiply $g^{3}$ by $g$ by adding the exponents.

Move $g$ .

$6 (y + 7) (g \cdot g^{3}) g \cdot g \cdot g \cdot g = 3 y$

Multiply $g$ by $g^{3}$ .

Raise $g$ to the power of $1$ .

$6 (y + 7) (g^{1} g^{3}) g \cdot g \cdot g \cdot g = 3 y$

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

$6 (y + 7) g^{1 + 3} g \cdot g \cdot g \cdot g = 3 y$

Add $1$ and $3$ .

$6 (y + 7) g^{4} g \cdot g \cdot g \cdot g = 3 y$

Multiply $g^{4}$ by $g$ by adding the exponents.

Move $g$ .

$6 (y + 7) (g \cdot g^{4}) g \cdot g \cdot g = 3 y$

Multiply $g$ by $g^{4}$ .

Raise $g$ to the power of $1$ .

$6 (y + 7) (g^{1} g^{4}) g \cdot g \cdot g = 3 y$

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

$6 (y + 7) g^{1 + 4} g \cdot g \cdot g = 3 y$

Add $1$ and $4$ .

$6 (y + 7) g^{5} g \cdot g \cdot g = 3 y$

Multiply $g^{5}$ by $g$ by adding the exponents.

Move $g$ .

$6 (y + 7) (g \cdot g^{5}) g \cdot g = 3 y$

Multiply $g$ by $g^{5}$ .

Raise $g$ to the power of $1$ .

$6 (y + 7) (g^{1} g^{5}) g \cdot g = 3 y$

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

$6 (y + 7) g^{1 + 5} g \cdot g = 3 y$

Add $1$ and $5$ .

$6 (y + 7) g^{6} g \cdot g = 3 y$

Multiply $g^{6}$ by $g$ by adding the exponents.

Move $g$ .

$6 (y + 7) (g \cdot g^{6}) g = 3 y$

Multiply $g$ by $g^{6}$ .

Raise $g$ to the power of $1$ .

$6 (y + 7) (g^{1} g^{6}) g = 3 y$

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

$6 (y + 7) g^{1 + 6} g = 3 y$

Add $1$ and $6$ .

$6 (y + 7) g^{7} g = 3 y$

Multiply $g^{7}$ by $g$ by adding the exponents.

Move $g$ .

$6 (y + 7) (g \cdot g^{7}) = 3 y$

Multiply $g$ by $g^{7}$ .

Raise $g$ to the power of $1$ .

$6 (y + 7) (g^{1} g^{7}) = 3 y$

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

$6 (y + 7) g^{1 + 7} = 3 y$

Add $1$ and $7$ .

$6 (y + 7) g^{8} = 3 y$

Apply the distributive property.

$(6 y + 6 \cdot 7) g^{8} = 3 y$

Multiply $6$ by $7$ .

$(6 y + 42) g^{8} = 3 y$

Apply the distributive property.

$6 y g^{8} + 42 g^{8} = 3 y$

Step 2

Subtract $3 y$ from both sides of the equation.

$6 y g^{8} + 42 g^{8} - 3 y = 0$

Step 3

Subtract $42 g^{8}$ from both sides of the equation.

$6 y g^{8} - 3 y = - 42 g^{8}$

Step 4

Factor $3 y$ out of $6 y g^{8} - 3 y$ .

Factor $3 y$ out of $6 y g^{8}$ .

$3 y (2 g^{8}) - 3 y = - 42 g^{8}$

Factor $3 y$ out of $- 3 y$ .

$3 y (2 g^{8}) + 3 y (- 1) = - 42 g^{8}$

Factor $3 y$ out of $3 y (2 g^{8}) + 3 y (- 1)$ .

$3 y (2 g^{8} - 1) = - 42 g^{8}$

Step 5

Divide each term in $3 y (2 g^{8} - 1) = - 42 g^{8}$ by $3 (2 g^{8} - 1)$ and simplify.

Divide each term in $3 y (2 g^{8} - 1) = - 42 g^{8}$ by $3 (2 g^{8} - 1)$ .

$\frac{3 y (2 g^{8} - 1)}{3 (2 g^{8} - 1)} = \frac{- 42 g^{8}}{3 (2 g^{8} - 1)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y (2 g^{8} - 1)}{3 (2 g^{8} - 1)} = \frac{- 42 g^{8}}{3 (2 g^{8} - 1)}$

Rewrite the expression.

$\frac{y (2 g^{8} - 1)}{2 g^{8} - 1} = \frac{- 42 g^{8}}{3 (2 g^{8} - 1)}$

Cancel the common factor of $2 g^{8} - 1$ .

Cancel the common factor.

$\frac{y (2 g^{8} - 1)}{2 g^{8} - 1} = \frac{- 42 g^{8}}{3 (2 g^{8} - 1)}$

Divide $y$ by $1$ .

$y = \frac{- 42 g^{8}}{3 (2 g^{8} - 1)}$

Simplify the right side.

Cancel the common factor of $- 42$ and $3$ .

Factor $3$ out of $- 42 g^{8}$ .

$y = \frac{3 (- 14 g^{8})}{3 (2 g^{8} - 1)}$

Cancel the common factors.

Cancel the common factor.

$y = \frac{3 (- 14 g^{8})}{3 (2 g^{8} - 1)}$

Rewrite the expression.

$y = \frac{- 14 g^{8}}{2 g^{8} - 1}$

Move the negative in front of the fraction.

$y = - \frac{14 g^{8}}{2 g^{8} - 1}$

Step 6

Set the denominator in $\frac{14 g^{8}}{2 g^{8} - 1}$ equal to $0$ to find where the expression is undefined.

$2 g^{8} - 1 = 0$

Step 7

Solve for $g$ .

Add $1$ to both sides of the equation.

$2 g^{8} = 1$

Divide each term in $2 g^{8} = 1$ by $2$ and simplify.

Divide each term in $2 g^{8} = 1$ by $2$ .

$\frac{2 g^{8}}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 g^{8}}{2} = \frac{1}{2}$

Divide $g^{8}$ by $1$ .

$g^{8} = \frac{1}{2}$

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

$g = \pm \sqrt[8]{\frac{1}{2}}$

Simplify $\pm \sqrt[8]{\frac{1}{2}}$ .

Rewrite $\sqrt[8]{\frac{1}{2}}$ as $\frac{\sqrt[8]{1}}{\sqrt[8]{2}}$ .

$g = \pm \frac{\sqrt[8]{1}}{\sqrt[8]{2}}$

Any root of $1$ is $1$ .

$g = \pm \frac{1}{\sqrt[8]{2}}$

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

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

$g = \frac{1}{\sqrt[8]{2}}$

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

$g = - \frac{1}{\sqrt[8]{2}}$

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

$g = \frac{1}{\sqrt[8]{2}}, - \frac{1}{\sqrt[8]{2}}$

Step 8

The domain is all values of $g$ that make the expression defined.

Interval Notation:

$(- \infty, - \frac{1}{\sqrt[8]{2}}) \cup (- \frac{1}{\sqrt[8]{2}}, \frac{1}{\sqrt[8]{2}}) \cup (\frac{1}{\sqrt[8]{2}}, \infty)$

Set-Builder Notation:

${g | g \neq \frac{1}{\sqrt[8]{2}}, - \frac{1}{\sqrt[8]{2}}}$

Step 9

Find the Domain 6(y+7)gggggggg=3y

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