Solve for x ( square root of x-27)^3=64

${(\sqrt{x - 27})}^{3} = 64$

Step 1

Simplify ${\sqrt{x - 27}}^{3}$ .

Rewrite ${\sqrt{x - 27}}^{3}$ as ${({(x - 27)}^{3})}^{\frac{1}{2}}$ .

$\sqrt{{(x - 27)}^{3}} = 64$

Factor out ${(x - 27)}^{2}$ .

$\sqrt{{(x - 27)}^{2} (x - 27)} = 64$

Pull terms out from under the radical.

$(x - 27) \sqrt{x - 27} = 64$

Apply the distributive property.

$x \sqrt{x - 27} - 27 \sqrt{x - 27} = 64$

Step 2

Solve for $\sqrt{x - 27}$ .

Factor $\sqrt{x - 27}$ out of $x \sqrt{x - 27} - 27 \sqrt{x - 27}$ .

Factor $\sqrt{x - 27}$ out of $x \sqrt{x - 27}$ .

$\sqrt{x - 27} x - 27 \sqrt{x - 27} = 64$

Factor $\sqrt{x - 27}$ out of $- 27 \sqrt{x - 27}$ .

$\sqrt{x - 27} x + \sqrt{x - 27} \cdot - 27 = 64$

Factor $\sqrt{x - 27}$ out of $\sqrt{x - 27} x + \sqrt{x - 27} \cdot - 27$ .

$\sqrt{x - 27} (x - 27) = 64$

Divide each term in $\sqrt{x - 27} (x - 27) = 64$ by $x - 27$ and simplify.

Divide each term in $\sqrt{x - 27} (x - 27) = 64$ by $x - 27$ .

$\frac{\sqrt{x - 27} (x - 27)}{x - 27} = \frac{64}{x - 27}$

Simplify the left side.

Cancel the common factor of $x - 27$ .

Cancel the common factor.

$\frac{\sqrt{x - 27} (x - 27)}{x - 27} = \frac{64}{x - 27}$

Divide $\sqrt{x - 27}$ by $1$ .

$\sqrt{x - 27} = \frac{64}{x - 27}$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x - 27}}^{2} = {(\frac{64}{x - 27})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 27}$ as ${(x - 27)}^{\frac{1}{2}}$ .

${({(x - 27)}^{\frac{1}{2}})}^{2} = {(\frac{64}{x - 27})}^{2}$

Simplify the left side.

Simplify ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x - 27)}^{\frac{1}{2}})}^{2}$ .

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

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(\frac{64}{x - 27})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 27)}^{\frac{1}{2} \cdot 2} = {(\frac{64}{x - 27})}^{2}$

Rewrite the expression.

${(x - 27)}^{1} = {(\frac{64}{x - 27})}^{2}$

Simplify.

$x - 27 = {(\frac{64}{x - 27})}^{2}$

Simplify the right side.

Simplify ${(\frac{64}{x - 27})}^{2}$ .

Apply the product rule to $\frac{64}{x - 27}$ .

$x - 27 = \frac{64^{2}}{{(x - 27)}^{2}}$

Raise $64$ to the power of $2$ .

$x - 27 = \frac{4096}{{(x - 27)}^{2}}$

Step 5

Solve for $x$ .

Add $27$ to both sides of the equation.

$x = \frac{4096}{{(x - 27)}^{2}} + 27$

Find the LCD of the terms in the equation.

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

$1, {(x - 27)}^{2}, 1$

The LCM of one and any expression is the expression.

${(x - 27)}^{2}$

Multiply each term in $x = \frac{4096}{{(x - 27)}^{2}} + 27$ by ${(x - 27)}^{2}$ to eliminate the fractions.

Multiply each term in $x = \frac{4096}{{(x - 27)}^{2}} + 27$ by ${(x - 27)}^{2}$ .

$x {(x - 27)}^{2} = \frac{4096}{{(x - 27)}^{2}} {(x - 27)}^{2} + 27 {(x - 27)}^{2}$

Simplify the right side.

Cancel the common factor of ${(x - 27)}^{2}$ .

Cancel the common factor.

$x {(x - 27)}^{2} = \frac{4096}{{(x - 27)}^{2}} {(x - 27)}^{2} + 27 {(x - 27)}^{2}$

Rewrite the expression.

$x {(x - 27)}^{2} = 4096 + 27 {(x - 27)}^{2}$

Solve the equation.

Simplify $4096 + 27 {(x - 27)}^{2}$ .

Simplify each term.

Rewrite ${(x - 27)}^{2}$ as $(x - 27) (x - 27)$ .

$x {(x - 27)}^{2} = 4096 + 27 ((x - 27) (x - 27))$

Expand $(x - 27) (x - 27)$ using the FOIL Method.

Apply the distributive property.

$x {(x - 27)}^{2} = 4096 + 27 (x (x - 27) - 27 (x - 27))$

Apply the distributive property.

$x {(x - 27)}^{2} = 4096 + 27 (x \cdot x + x \cdot - 27 - 27 (x - 27))$

Apply the distributive property.

$x {(x - 27)}^{2} = 4096 + 27 (x \cdot x + x \cdot - 27 - 27 x - 27 \cdot - 27)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x {(x - 27)}^{2} = 4096 + 27 (x^{2} + x \cdot - 27 - 27 x - 27 \cdot - 27)$

Move $- 27$ to the left of $x$ .

$x {(x - 27)}^{2} = 4096 + 27 (x^{2} - 27 \cdot x - 27 x - 27 \cdot - 27)$

Multiply $- 27$ by $- 27$ .

$x {(x - 27)}^{2} = 4096 + 27 (x^{2} - 27 x - 27 x + 729)$

Subtract $27 x$ from $- 27 x$ .

$x {(x - 27)}^{2} = 4096 + 27 (x^{2} - 54 x + 729)$

Apply the distributive property.

$x {(x - 27)}^{2} = 4096 + 27 x^{2} + 27 (- 54 x) + 27 \cdot 729$

Simplify.

Multiply $- 54$ by $27$ .

$x {(x - 27)}^{2} = 4096 + 27 x^{2} - 1458 x + 27 \cdot 729$

Multiply $27$ by $729$ .

$x {(x - 27)}^{2} = 4096 + 27 x^{2} - 1458 x + 19683$

Add $4096$ and $19683$ .

$x {(x - 27)}^{2} = 27 x^{2} - 1458 x + 23779$

Move all terms containing $x$ to the left side of the equation.

Subtract $27 x^{2}$ from both sides of the equation.

$x {(x - 27)}^{2} - 27 x^{2} = - 1458 x + 23779$

Add $1458 x$ to both sides of the equation.

$x {(x - 27)}^{2} - 27 x^{2} + 1458 x = 23779$

Simplify each term.

Rewrite ${(x - 27)}^{2}$ as $(x - 27) (x - 27)$ .

$x ((x - 27) (x - 27)) - 27 x^{2} + 1458 x = 23779$

Expand $(x - 27) (x - 27)$ using the FOIL Method.

Apply the distributive property.

$x (x (x - 27) - 27 (x - 27)) - 27 x^{2} + 1458 x = 23779$

Apply the distributive property.

$x (x \cdot x + x \cdot - 27 - 27 (x - 27)) - 27 x^{2} + 1458 x = 23779$

Apply the distributive property.

$x (x \cdot x + x \cdot - 27 - 27 x - 27 \cdot - 27) - 27 x^{2} + 1458 x = 23779$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x (x^{2} + x \cdot - 27 - 27 x - 27 \cdot - 27) - 27 x^{2} + 1458 x = 23779$

Move $- 27$ to the left of $x$ .

$x (x^{2} - 27 \cdot x - 27 x - 27 \cdot - 27) - 27 x^{2} + 1458 x = 23779$

Multiply $- 27$ by $- 27$ .

$x (x^{2} - 27 x - 27 x + 729) - 27 x^{2} + 1458 x = 23779$

Subtract $27 x$ from $- 27 x$ .

$x (x^{2} - 54 x + 729) - 27 x^{2} + 1458 x = 23779$

Apply the distributive property.

$x \cdot x^{2} + x (- 54 x) + x \cdot 729 - 27 x^{2} + 1458 x = 23779$

Simplify.

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

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

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

$x^{1} x^{2} + x (- 54 x) + x \cdot 729 - 27 x^{2} + 1458 x = 23779$

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

$x^{1 + 2} + x (- 54 x) + x \cdot 729 - 27 x^{2} + 1458 x = 23779$

Add $1$ and $2$ .

$x^{3} + x (- 54 x) + x \cdot 729 - 27 x^{2} + 1458 x = 23779$

Rewrite using the commutative property of multiplication.

$x^{3} - 54 x \cdot x + x \cdot 729 - 27 x^{2} + 1458 x = 23779$

Move $729$ to the left of $x$ .

$x^{3} - 54 x \cdot x + 729 \cdot x - 27 x^{2} + 1458 x = 23779$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{3} - 54 (x \cdot x) + 729 \cdot x - 27 x^{2} + 1458 x = 23779$

Multiply $x$ by $x$ .

$x^{3} - 54 x^{2} + 729 \cdot x - 27 x^{2} + 1458 x = 23779$

$x^{3} - 54 x^{2} + 729 x - 27 x^{2} + 1458 x = 23779$

Subtract $27 x^{2}$ from $- 54 x^{2}$ .

$x^{3} - 81 x^{2} + 729 x + 1458 x = 23779$

Add $729 x$ and $1458 x$ .

$x^{3} - 81 x^{2} + 2187 x = 23779$

Subtract $23779$ from both sides of the equation.

$x^{3} - 81 x^{2} + 2187 x - 23779 = 0$

Factor $x^{3} - 81 x^{2} + 2187 x - 23779$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 23779, \pm 7, \pm 3397, \pm 43, \pm 553, \pm 79, \pm 301 q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 23779, \pm 7, \pm 3397, \pm 43, \pm 553, \pm 79, \pm 301$

Substitute $43$ and simplify the expression. In this case, the expression is equal to $0$ so $43$ is a root of the polynomial.

Substitute $43$ into the polynomial.

$43^{3} - 81 \cdot 43^{2} + 2187 \cdot 43 - 23779$

Raise $43$ to the power of $3$ .

$79507 - 81 \cdot 43^{2} + 2187 \cdot 43 - 23779$

Raise $43$ to the power of $2$ .

$79507 - 81 \cdot 1849 + 2187 \cdot 43 - 23779$

Multiply $- 81$ by $1849$ .

$79507 - 149769 + 2187 \cdot 43 - 23779$

Subtract $149769$ from $79507$ .

$- 70262 + 2187 \cdot 43 - 23779$

Multiply $2187$ by $43$ .

$- 70262 + 94041 - 23779$

Add $- 70262$ and $94041$ .

$23779 - 23779$

Subtract $23779$ from $23779$ .

$0$

Since $43$ is a known root, divide the polynomial by $x - 43$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 81 x^{2} + 2187 x - 23779}{x - 43}$

Divide $x^{3} - 81 x^{2} + 2187 x - 23779$ by $x - 43$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$43$

$x^{3}$

$81 x^{2}$

$2187 x$

$23779$

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			+	$x^{3}$	-	$43 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 43 x^{2}$

				$x^{2}$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$

Divide the highest order term in the dividend $- 38 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$38 x$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$38 x$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					-	$38 x^{2}$	+	$1634 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 38 x^{2} + 1634 x$

				$x^{2}$	-	$38 x$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					+	$38 x^{2}$	-	$1634 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$38 x$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					+	$38 x^{2}$	-	$1634 x$
							+	$553 x$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$38 x$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					+	$38 x^{2}$	-	$1634 x$
							+	$553 x$	-	$23779$

Divide the highest order term in the dividend $553 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$38 x$	+	$553$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					+	$38 x^{2}$	-	$1634 x$
							+	$553 x$	-	$23779$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$38 x$	+	$553$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					+	$38 x^{2}$	-	$1634 x$
							+	$553 x$	-	$23779$
							+	$553 x$	-	$23779$

The expression needs to be subtracted from the dividend, so change all the signs in $553 x - 23779$

				$x^{2}$	-	$38 x$	+	$553$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					+	$38 x^{2}$	-	$1634 x$
							+	$553 x$	-	$23779$
							-	$553 x$	+	$23779$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$38 x$	+	$553$
$x$	-	$43$		$x^{3}$	-	$81 x^{2}$	+	$2187 x$	-	$23779$
			-	$x^{3}$	+	$43 x^{2}$
					-	$38 x^{2}$	+	$2187 x$
					+	$38 x^{2}$	-	$1634 x$
							+	$553 x$	-	$23779$
							-	$553 x$	+	$23779$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 38 x + 553$

Write $x^{3} - 81 x^{2} + 2187 x - 23779$ as a set of factors.

$(x - 43) (x^{2} - 38 x + 553) = 0$

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

$x - 43 = 0$

$x^{2} - 38 x + 553 = 0$

Set $x - 43$ equal to $0$ and solve for $x$ .

Set $x - 43$ equal to $0$ .

$x - 43 = 0$

Add $43$ to both sides of the equation.

$x = 43$

Set $x^{2} - 38 x + 553$ equal to $0$ and solve for $x$ .

Set $x^{2} - 38 x + 553$ equal to $0$ .

$x^{2} - 38 x + 553 = 0$

Solve $x^{2} - 38 x + 553 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 38$ , and $c = 553$ into the quadratic formula and solve for $x$ .

$\frac{38 \pm \sqrt{{(- 38)}^{2} - 4 \cdot (1 \cdot 553)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 38$ to the power of $2$ .

$x = \frac{38 \pm \sqrt{1444 - 4 \cdot 1 \cdot 553}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 553$ .

Multiply $- 4$ by $1$ .

$x = \frac{38 \pm \sqrt{1444 - 4 \cdot 553}}{2 \cdot 1}$

Multiply $- 4$ by $553$ .

$x = \frac{38 \pm \sqrt{1444 - 2212}}{2 \cdot 1}$

Subtract $2212$ from $1444$ .

$x = \frac{38 \pm \sqrt{- 768}}{2 \cdot 1}$

Rewrite $- 768$ as $- 1 (768)$ .

$x = \frac{38 \pm \sqrt{- 1 \cdot 768}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (768)}$ as $\sqrt{- 1} \cdot \sqrt{768}$ .

$x = \frac{38 \pm \sqrt{- 1} \cdot \sqrt{768}}{2 \cdot 1}$

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

$x = \frac{38 \pm i \cdot \sqrt{768}}{2 \cdot 1}$

Rewrite $768$ as $16^{2} \cdot 3$ .

Factor $256$ out of $768$ .

$x = \frac{38 \pm i \cdot \sqrt{256 (3)}}{2 \cdot 1}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{38 \pm i \cdot \sqrt{16^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{38 \pm i \cdot (16 \sqrt{3})}{2 \cdot 1}$

Move $16$ to the left of $i$ .

$x = \frac{38 \pm 16 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{38 \pm 16 i \sqrt{3}}{2}$

Simplify $\frac{38 \pm 16 i \sqrt{3}}{2}$ .

$x = 19 \pm 8 i \sqrt{3}$

The final answer is the combination of both solutions.

$x = 19 + 8 i \sqrt{3}, 19 - 8 i \sqrt{3}$

The final solution is all the values that make $(x - 43) (x^{2} - 38 x + 553) = 0$ true.

$x = 43, 19 + 8 i \sqrt{3}, 19 - 8 i \sqrt{3}$

Solve for x ( square root of x-27)^3=64

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