Solve by Factoring x^-3=1/64

$x^{- 3} = \frac{1}{64}$

Move $\frac{1}{64}$ to the left side of the equation by subtracting it from both sides.

$x^{- 3} - \frac{1}{64} = 0$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{3}} - \frac{1}{64} = 0$

Rewrite $1$ as $1^{3}$ .

$\frac{1^{3}}{x^{3}} - \frac{1}{64} = 0$

Rewrite $\frac{1^{3}}{x^{3}}$ as ${(\frac{1}{x})}^{3}$ .

${(\frac{1}{x})}^{3} - \frac{1}{64} = 0$

Rewrite $\frac{1}{64}$ as ${(\frac{1}{4})}^{3}$ .

${(\frac{1}{x})}^{3} - {(\frac{1}{4})}^{3} = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = \frac{1}{x}$ and $b = \frac{1}{4}$ .

$(\frac{1}{x} - \frac{1}{4}) ({(\frac{1}{x})}^{2} + \frac{1}{x} \cdot \frac{1}{4} + {(\frac{1}{4})}^{2}) = 0$

Simplify.

Tap for more steps…

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

$(\frac{1}{x} - \frac{1}{4}) (\frac{1^{2}}{x^{2}} + \frac{1}{x} \cdot \frac{1}{4} + {(\frac{1}{4})}^{2}) = 0$

One to any power is one.

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{x} \cdot \frac{1}{4} + {(\frac{1}{4})}^{2}) = 0$

Multiply $\frac{1}{x}$ and $\frac{1}{4}$ .

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{x \cdot 4} + {(\frac{1}{4})}^{2}) = 0$

Move $4$ to the left of $x$ .

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{4 \cdot x} + {(\frac{1}{4})}^{2}) = 0$

Multiply $4$ by $x$ .

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{4 x} + {(\frac{1}{4})}^{2}) = 0$

Apply the product rule to $\frac{1}{4}$ .

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{4 x} + \frac{1^{2}}{4^{2}}) = 0$

One to any power is one.

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{4 x} + \frac{1}{4^{2}}) = 0$

Raise $4$ to the power of $2$ .

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{x^{2}} + \frac{1}{4 x} + \frac{1}{16}) = 0$

Reorder terms.

$(\frac{1}{x} - \frac{1}{4}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(\frac{1}{x} \cdot \frac{4}{4} - \frac{1}{4}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$(\frac{1}{x} \cdot \frac{4}{4} - \frac{1}{4} \cdot \frac{x}{x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Write each expression with a common denominator of $x \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Multiply $\frac{1}{x}$ and $\frac{4}{4}$ .

$(\frac{4}{x \cdot 4} - \frac{1}{4} \cdot \frac{x}{x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Multiply $\frac{1}{4}$ and $\frac{x}{x}$ .

$(\frac{4}{x \cdot 4} - \frac{x}{4 x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Reorder the factors of $x \cdot 4$ .

$(\frac{4}{4 x} - \frac{x}{4 x}) (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Combine the numerators over the common denominator.

$\frac{4 - x}{4 x} \cdot (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

Reorder terms.

$\frac{- x + 4}{4 x} \cdot (\frac{1}{4 x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

To write $\frac{1}{4 x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{1}{4 x} \cdot \frac{x}{x} + \frac{1}{x^{2}} + \frac{1}{16}) = 0$

To write $\frac{1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{1}{4 x} \cdot \frac{x}{x} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Write each expression with a common denominator of $4 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Multiply $\frac{1}{4 x}$ and $\frac{x}{x}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{x}{4 x \cdot x} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

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

$\frac{- x + 4}{4 x} \cdot (\frac{x}{4 (x \cdot x)} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

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

$\frac{- x + 4}{4 x} \cdot (\frac{x}{4 (x \cdot x)} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

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

$\frac{- x + 4}{4 x} \cdot (\frac{x}{4 x^{1 + 1}} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Add $1$ and $1$ .

$\frac{- x + 4}{4 x} \cdot (\frac{x}{4 x^{2}} + \frac{1}{x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

Multiply $\frac{1}{x^{2}}$ and $\frac{4}{4}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{x}{4 x^{2}} + \frac{4}{x^{2} \cdot 4} + \frac{1}{16}) = 0$

Reorder the factors of $x^{2} \cdot 4$ .

$\frac{- x + 4}{4 x} \cdot (\frac{x}{4 x^{2}} + \frac{4}{4 x^{2}} + \frac{1}{16}) = 0$

Combine the numerators over the common denominator.

$\frac{- x + 4}{4 x} \cdot (\frac{x + 4}{4 x^{2}} + \frac{1}{16}) = 0$

To write $\frac{x + 4}{4 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{x + 4}{4 x^{2}} \cdot \frac{4}{4} + \frac{1}{16}) = 0$

To write $\frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{x + 4}{4 x^{2}} \cdot \frac{4}{4} + \frac{1}{16} \cdot \frac{x^{2}}{x^{2}}) = 0$

Write each expression with a common denominator of $16 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps…

Multiply $\frac{x + 4}{4 x^{2}}$ and $\frac{4}{4}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{(x + 4) \cdot 4}{4 x^{2} \cdot 4} + \frac{1}{16} \cdot \frac{x^{2}}{x^{2}}) = 0$

Multiply $4$ by $4$ .

$\frac{- x + 4}{4 x} \cdot (\frac{(x + 4) \cdot 4}{16 x^{2}} + \frac{1}{16} \cdot \frac{x^{2}}{x^{2}}) = 0$

Multiply $\frac{1}{16}$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{- x + 4}{4 x} \cdot (\frac{(x + 4) \cdot 4}{16 x^{2}} + \frac{x^{2}}{16 x^{2}}) = 0$

Combine the numerators over the common denominator.

$\frac{- x + 4}{4 x} \cdot \frac{(x + 4) \cdot 4 + x^{2}}{16 x^{2}} = 0$

Reorder terms.

$\frac{- x + 4}{4 x} \cdot \frac{x^{2} + 4 (x + 4)}{16 x^{2}} = 0$

Rewrite $\frac{x^{2} + 4 (x + 4)}{16 x^{2}}$ in a factored form.

Tap for more steps…

Apply the distributive property.

$\frac{- x + 4}{4 x} \cdot \frac{x^{2} + 4 x + 4 \cdot 4}{16 x^{2}} = 0$

Multiply $4$ by $4$ .

$\frac{- x + 4}{4 x} \cdot \frac{x^{2} + 4 x + 16}{16 x^{2}} = 0$

Factor each term.

Tap for more steps…

Multiply $\frac{- x + 4}{4 x}$ and $\frac{x^{2} + 4 x + 16}{16 x^{2}}$ .

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{4 x (16 x^{2})} = 0$

Multiply $16$ by $4$ .

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x \cdot x^{2}} = 0$

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

Tap for more steps…

Move $x^{2}$ .

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 (x^{2} x)} = 0$

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

Tap for more steps…

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

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 (x^{2} x^{1})} = 0$

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

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x^{2 + 1}} = 0$

Add $2$ and $1$ .

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x^{3}} = 0$

Find the LCD of the terms in the equation.

Tap for more steps…

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$64 x^{3}, 1$

Since $64 x^{3}, 1$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $64, 1$ then find LCM for the variable part $x^{3}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The prime factors for $64$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ .

Tap for more steps…

$64$ has factors of $2$ and $32$ .

$2 \cdot 32$

$32$ has factors of $2$ and $16$ .

$2 \cdot 2 \cdot 16$

$16$ has factors of $2$ and $8$ .

$2 \cdot 2 \cdot 2 \cdot 8$

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 4$

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $64, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

The LCM of $64, 1$ is $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64$ .

Tap for more steps…

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 2 \cdot 2$

Multiply $8$ by $2$ .

$16 \cdot 2 \cdot 2$

Multiply $16$ by $2$ .

$32 \cdot 2$

Multiply $32$ by $2$ .

$64$

The factors for $x^{3}$ are $x \cdot x \cdot x$ , which is $x$ multiplied by each other $3$ times.

$x^{3} = x \cdot x \cdot x$

$x$ occurs $3$ times.

The LCM of $x^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x$

Simplify $x \cdot x \cdot x$ .

Tap for more steps…

Multiply $x$ by $x$ .

$x^{2} \cdot x$

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

Tap for more steps…

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

Tap for more steps…

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

$x^{2} \cdot x^{1}$

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

$x^{2 + 1}$

Add $2$ and $1$ .

$x^{3}$

The LCM for $64 x^{3}, 1$ is the numeric part $64$ multiplied by the variable part.

$64 x^{3}$

Multiply each term by $64 x^{3}$ and simplify.

Tap for more steps…

Multiply each term in $\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x^{3}} = 0$ by $64 x^{3}$ in order to remove all the denominators from the equation.

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x^{3}} \cdot (64 x^{3}) = 0 \cdot (64 x^{3})$

Simplify $\frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x^{3}} \cdot (64 x^{3})$ .

Tap for more steps…

Reduce the expression by cancelling the common factors.

Tap for more steps…

Rewrite using the commutative property of multiplication.

$64 \frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x^{3}} x^{3} = 0 \cdot (64 x^{3})$

Cancel the common factor of $64$ .

Tap for more steps…

Factor $64$ out of $64 x^{3}$ .

$64 \frac{(- x + 4) (x^{2} + 4 x + 16)}{64 (x^{3})} x^{3} = 0 \cdot (64 x^{3})$

Cancel the common factor.

$64 \frac{(- x + 4) (x^{2} + 4 x + 16)}{64 x^{3}} x^{3} = 0 \cdot (64 x^{3})$

Rewrite the expression.

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{x^{3}} x^{3} = 0 \cdot (64 x^{3})$

Cancel the common factor of $x^{3}$ .

Tap for more steps…

Cancel the common factor.

$\frac{(- x + 4) (x^{2} + 4 x + 16)}{x^{3}} x^{3} = 0 \cdot (64 x^{3})$

Rewrite the expression.

$(- x + 4) (x^{2} + 4 x + 16) = 0 \cdot (64 x^{3})$

Expand $(- x + 4) (x^{2} + 4 x + 16)$ by multiplying each term in the first expression by each term in the second expression.

$- x \cdot x^{2} - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

Simplify terms.

Tap for more steps…

Simplify each term.

Tap for more steps…

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

Tap for more steps…

Move $x^{2}$ .

$- (x^{2} x) - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

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

Tap for more steps…

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

$- (x^{2} x^{1}) - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

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

$- x^{2 + 1} - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

Add $2$ and $1$ .

$- x^{3} - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

Multiply $x$ by $x$ .

$- x^{3} - 1 \cdot 4 x^{2} - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

Multiply $- 1$ by $4$ .

$- x^{3} - 4 x^{2} - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

Multiply $16$ by $- 1$ .

$- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 4 (4 x) + 4 \cdot 16 = 0 \cdot (64 x^{3})$

Multiply $4$ by $4$ .

$- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 16 x + 4 \cdot 16 = 0 \cdot (64 x^{3})$

Multiply $4$ by $16$ .

$- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 16 x + 64 = 0 \cdot (64 x^{3})$

Combine the opposite terms in $- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 16 x + 64$ .

Tap for more steps…

Add $- 4 x^{2}$ and $4 x^{2}$ .

$- x^{3} - 16 x + 0 + 16 x + 64 = 0 \cdot (64 x^{3})$

Add $- x^{3} - 16 x$ and $0$ .

$- x^{3} - 16 x + 16 x + 64 = 0 \cdot (64 x^{3})$

Add $- 16 x$ and $16 x$ .

$- x^{3} + 0 + 64 = 0 \cdot (64 x^{3})$

Add $- x^{3}$ and $0$ .

$- x^{3} + 64 = 0 \cdot (64 x^{3})$

Multiply $0 (64 x^{3})$ .

Tap for more steps…

Multiply $64$ by $0$ .

$- x^{3} + 64 = 0 x^{3}$

Multiply $0$ by $x^{3}$ .

$- x^{3} + 64 = 0$

Solve the equation.

Tap for more steps…

Subtract $64$ from both sides of the equation.

$- x^{3} = - 64$

Move $64$ to the left side of the equation by adding it to both sides.

$- x^{3} + 64 = 0$

Factor the left side of the equation.

Tap for more steps…

Factor $- 1$ out of $- x^{3} + 64$ .

Tap for more steps…

Factor $- 1$ out of $- x^{3}$ .

$- (x^{3}) + 64 = 0$

Rewrite $64$ as $- 1 (- 64)$ .

$- (x^{3}) - 1 \cdot - 64 = 0$

Factor $- 1$ out of $- (x^{3}) - 1 (- 64)$ .

$- (x^{3} - 64) = 0$

Rewrite $64$ as $4^{3}$ .

$- (x^{3} - 4^{3}) = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

$- ((x - 4) (x^{2} + x \cdot 4 + 4^{2})) = 0$

Factor.

Tap for more steps…

Simplify.

Tap for more steps…

Move $4$ to the left of $x$ .

$- ((x - 4) (x^{2} + 4 x + 4^{2})) = 0$

Raise $4$ to the power of $2$ .

$- ((x - 4) (x^{2} + 4 x + 16)) = 0$

Remove unnecessary parentheses.

$- (x - 4) (x^{2} + 4 x + 16) = 0$

Multiply each term in $- (x - 4) (x^{2} + 4 x + 16) = 0$ by $- 1$

Tap for more steps…

Multiply each term in $- (x - 4) (x^{2} + 4 x + 16) = 0$ by $- 1$ .

$(- (x - 4) (x^{2} + 4 x + 16)) \cdot - 1 = 0 \cdot - 1$

Simplify $(- (x - 4) (x^{2} + 4 x + 16)) \cdot - 1$ .

Tap for more steps…

Simplify by multiplying through.

Tap for more steps…

Apply the distributive property.

$(- x - - 4) (x^{2} + 4 x + 16) \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $- 4$ .

$(- x + 4) (x^{2} + 4 x + 16) \cdot - 1 = 0 \cdot - 1$

Expand $(- x + 4) (x^{2} + 4 x + 16)$ by multiplying each term in the first expression by each term in the second expression.

$(- x \cdot x^{2} - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

Simplify terms.

Tap for more steps…

Simplify each term.

Tap for more steps…

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

Tap for more steps…

Move $x^{2}$ .

$(- (x^{2} x) - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

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

Tap for more steps…

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

$(- (x^{2} x^{1}) - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

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

$(- x^{2 + 1} - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

Add $2$ and $1$ .

$(- x^{3} - x (4 x) - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

Multiply $x$ by $x$ .

$(- x^{3} - 1 \cdot 4 x^{2} - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $4$ .

$(- x^{3} - 4 x^{2} - x \cdot 16 + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

Multiply $16$ by $- 1$ .

$(- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 4 (4 x) + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

Multiply $4$ by $4$ .

$(- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 16 x + 4 \cdot 16) \cdot - 1 = 0 \cdot - 1$

Multiply $4$ by $16$ .

$(- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 16 x + 64) \cdot - 1 = 0 \cdot - 1$

Simplify terms.

Tap for more steps…

Combine the opposite terms in $- x^{3} - 4 x^{2} - 16 x + 4 x^{2} + 16 x + 64$ .

Tap for more steps…

Add $- 4 x^{2}$ and $4 x^{2}$ .

$(- x^{3} - 16 x + 0 + 16 x + 64) \cdot - 1 = 0 \cdot - 1$

Add $- x^{3} - 16 x$ and $0$ .

$(- x^{3} - 16 x + 16 x + 64) \cdot - 1 = 0 \cdot - 1$

Add $- 16 x$ and $16 x$ .

$(- x^{3} + 0 + 64) \cdot - 1 = 0 \cdot - 1$

Add $- x^{3}$ and $0$ .

$(- x^{3} + 64) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$- x^{3} \cdot - 1 + 64 \cdot - 1 = 0 \cdot - 1$

Multiply $- x^{3} \cdot - 1$ .

Tap for more steps…

Multiply $- 1$ by $- 1$ .

$1 x^{3} + 64 \cdot - 1 = 0 \cdot - 1$

Multiply $x^{3}$ by $1$ .

$x^{3} + 64 \cdot - 1 = 0 \cdot - 1$

Multiply $64$ by $- 1$ .

$x^{3} - 64 = 0 \cdot - 1$

Multiply $0$ by $- 1$ .

$x^{3} - 64 = 0$

Add $64$ to both sides of the equation.

$x^{3} = 64$

Move $64$ to the left side of the equation by subtracting it from both sides.

$x^{3} - 64 = 0$

Factor the left side of the equation.

Tap for more steps…

Rewrite $64$ as $4^{3}$ .

$x^{3} - 4^{3} = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

$(x - 4) (x^{2} + x \cdot 4 + 4^{2}) = 0$

Simplify.

Tap for more steps…

Move $4$ to the left of $x$ .

$(x - 4) (x^{2} + 4 x + 4^{2}) = 0$

Raise $4$ to the power of $2$ .

$(x - 4) (x^{2} + 4 x + 16) = 0$

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

$x - 4 = 0$

$x^{2} + 4 x + 16 = 0$

Set the first factor equal to $0$ and solve.

Tap for more steps…

Set the first factor equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set the next factor equal to $0$ and solve.

Tap for more steps…

Set the next factor equal to $0$ .

$x^{2} + 4 x + 16 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Simplify.

Tap for more steps…

Simplify the numerator.

Tap for more steps…

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Multiply $16$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps…

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps…

Simplify the numerator.

Tap for more steps…

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Multiply $16$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps…

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Change the $\pm$ to $+$ .

$x = - 2 + 2 i \sqrt{3}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps…

Simplify the numerator.

Tap for more steps…

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Multiply $16$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps…

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Change the $\pm$ to $-$ .

$x = - 2 - 2 i \sqrt{3}$

The final answer is the combination of both solutions.

$x = - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

The final solution is all the values that make $(x - 4) (x^{2} + 4 x + 16) = 0$ true.

$x = 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Solve by Factoring x^-3=1/64