Algebra Examples

$5 y^{2} \cdot {(2 y^{3})}^{x} = 80 y^{14}$

Step 1

Divide each term in $5 y^{2} \cdot {(2 y^{3})}^{x} = 80 y^{14}$ by $5 y^{2}$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $5 y^{2} \cdot {(2 y^{3})}^{x} = 80 y^{14}$ by $5 y^{2}$ .

$\frac{5 y^{2} \cdot {(2 y^{3})}^{x}}{5 y^{2}} = \frac{80 y^{14}}{5 y^{2}}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{5 y^{2} \cdot {(2 y^{3})}^{x}}{5 y^{2}} = \frac{80 y^{14}}{5 y^{2}}$

Step 1.2.1.2

Rewrite the expression.

$\frac{y^{2} \cdot {(2 y^{3})}^{x}}{y^{2}} = \frac{80 y^{14}}{5 y^{2}}$

Step 1.2.2

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 1.2.2.1

Cancel the common factor.

$\frac{y^{2} \cdot {(2 y^{3})}^{x}}{y^{2}} = \frac{80 y^{14}}{5 y^{2}}$

Step 1.2.2.2

Divide ${(2 y^{3})}^{x}$ by $1$ .

${(2 y^{3})}^{x} = \frac{80 y^{14}}{5 y^{2}}$

Step 1.2.3

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.3.1

Apply the product rule to $2 y^{3}$ .

$2^{x} {(y^{3})}^{x} = \frac{80 y^{14}}{5 y^{2}}$

Step 1.2.3.2

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

$2^{x} y^{3 x} = \frac{80 y^{14}}{5 y^{2}}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Cancel the common factor of $80$ and $5$ .

Tap for more steps...

Step 1.3.1.1

Factor $5$ out of $80 y^{14}$ .

$2^{x} y^{3 x} = \frac{5 (16 y^{14})}{5 y^{2}}$

Step 1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.3.1.2.1

Factor $5$ out of $5 y^{2}$ .

$2^{x} y^{3 x} = \frac{5 (16 y^{14})}{5 (y^{2})}$

Step 1.3.1.2.2

Cancel the common factor.

$2^{x} y^{3 x} = \frac{5 (16 y^{14})}{5 y^{2}}$

Step 1.3.1.2.3

Rewrite the expression.

$2^{x} y^{3 x} = \frac{16 y^{14}}{y^{2}}$

Step 1.3.2

Cancel the common factor of $y^{14}$ and $y^{2}$ .

Tap for more steps...

Step 1.3.2.1

Factor $y^{2}$ out of $16 y^{14}$ .

$2^{x} y^{3 x} = \frac{y^{2} (16 y^{12})}{y^{2}}$

Step 1.3.2.2

Cancel the common factors.

Tap for more steps...

Step 1.3.2.2.1

Multiply by $1$ .

$2^{x} y^{3 x} = \frac{y^{2} (16 y^{12})}{y^{2} \cdot 1}$

Step 1.3.2.2.2

Cancel the common factor.

$2^{x} y^{3 x} = \frac{y^{2} (16 y^{12})}{y^{2} \cdot 1}$

Step 1.3.2.2.3

Rewrite the expression.

$2^{x} y^{3 x} = \frac{16 y^{12}}{1}$

Step 1.3.2.2.4

Divide $16 y^{12}$ by $1$ .

$2^{x} y^{3 x} = 16 y^{12}$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (2^{x} y^{3 x}) = \ln (16 y^{12})$

Step 3

Expand the left side.

Tap for more steps...

Step 3.1

Rewrite $\ln (2^{x} y^{3 x})$ as $\ln (2^{x}) + \ln (y^{3 x})$ .

$\ln (2^{x}) + \ln (y^{3 x}) = \ln (16 y^{12})$

Step 3.2

Expand $\ln (2^{x})$ by moving $x$ outside the logarithm.

$x \ln (2) + \ln (y^{3 x}) = \ln (16 y^{12})$

Step 3.3

Expand $\ln (y^{3 x})$ by moving $3 x$ outside the logarithm.

$x \ln (2) + 3 x \ln (y) = \ln (16 y^{12})$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (2) + 3 x \ln (y) - \ln (16 y^{12}) = 0$

Step 5

Add $\ln (16 y^{12})$ to both sides of the equation.

$x \ln (2) + 3 x \ln (y) = \ln (16 y^{12})$

Step 6

Factor $x$ out of $x \ln (2) + 3 x \ln (y)$ .

Tap for more steps...

Step 6.1

Factor $x$ out of $x \ln (2)$ .

$x (\ln (2)) + 3 x \ln (y) = \ln (16 y^{12})$

Step 6.2

Factor $x$ out of $3 x \ln (y)$ .

$x (\ln (2)) + x (3 \ln (y)) = \ln (16 y^{12})$

Step 6.3

Factor $x$ out of $x (\ln (2)) + x (3 \ln (y))$ .

$x (\ln (2) + 3 \ln (y)) = \ln (16 y^{12})$

Step 7

Divide each term in $x (\ln (2) + 3 \ln (y)) = \ln (16 y^{12})$ by $\ln (2) + 3 \ln (y)$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $x (\ln (2) + 3 \ln (y)) = \ln (16 y^{12})$ by $\ln (2) + 3 \ln (y)$ .

$\frac{x (\ln (2) + 3 \ln (y))}{\ln (2) + 3 \ln (y)} = \frac{\ln (16 y^{12})}{\ln (2) + 3 \ln (y)}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $\ln (2) + 3 \ln (y)$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{x (\ln (2) + 3 \ln (y))}{\ln (2) + 3 \ln (y)} = \frac{\ln (16 y^{12})}{\ln (2) + 3 \ln (y)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (16 y^{12})}{\ln (2) + 3 \ln (y)}$