Algebra Examples

${(x + 1)}^{\frac{3}{2}} - 2 = 25$

Step 1

Subtract $25$ from both sides of the equation.

${(x + 1)}^{\frac{3}{2}} - 2 - 25 = 0$

Step 2

Subtract $25$ from $- 2$ .

${(x + 1)}^{\frac{3}{2}} - 27 = 0$

Step 3

Rewrite ${(x + 1)}^{\frac{3}{2}}$ as ${({(x + 1)}^{\frac{1}{2}})}^{3}$ .

${({(x + 1)}^{\frac{1}{2}})}^{3} - 27 = 0$

Step 4

Rewrite $27$ as $3^{3}$ .

${({(x + 1)}^{\frac{1}{2}})}^{3} - 3^{3} = 0$

Step 5

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 $a = {(x + 1)}^{\frac{1}{2}}$ and $b = 3$ .

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

Step 6

Simplify.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Cancel the common factor of $2$ .

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Step 6.1.2.1

Cancel the common factor.

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

Step 6.1.2.2

Rewrite the expression.

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

Step 6.2

Simplify.

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

Step 6.3

Move $3$ to the left of ${(x + 1)}^{\frac{1}{2}}$ .

$({(x + 1)}^{\frac{1}{2}} - 3) (x + 1 + 3 {(x + 1)}^{\frac{1}{2}} + 3^{2}) = 0$

Step 6.4

Raise $3$ to the power of $2$ .

$({(x + 1)}^{\frac{1}{2}} - 3) (x + 1 + 3 {(x + 1)}^{\frac{1}{2}} + 9) = 0$

Step 6.5

Add $1$ and $9$ .

$({(x + 1)}^{\frac{1}{2}} - 3) (x + 3 {(x + 1)}^{\frac{1}{2}} + 10) = 0$

Step 7

Simplify $({(x + 1)}^{\frac{1}{2}} - 3) (x + 3 {(x + 1)}^{\frac{1}{2}} + 10)$ .

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Step 7.1

Expand $({(x + 1)}^{\frac{1}{2}} - 3) (x + 3 {(x + 1)}^{\frac{1}{2}} + 10)$ by multiplying each term in the first expression by each term in the second expression.

${(x + 1)}^{\frac{1}{2}} x + {(x + 1)}^{\frac{1}{2}} (3 {(x + 1)}^{\frac{1}{2}}) + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2

Simplify terms.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Rewrite using the commutative property of multiplication.

${(x + 1)}^{\frac{1}{2}} x + 3 {(x + 1)}^{\frac{1}{2}} {(x + 1)}^{\frac{1}{2}} + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.2

Multiply ${(x + 1)}^{\frac{1}{2}}$ by ${(x + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 7.2.1.2.1

Move ${(x + 1)}^{\frac{1}{2}}$ .

${(x + 1)}^{\frac{1}{2}} x + 3 ({(x + 1)}^{\frac{1}{2}} {(x + 1)}^{\frac{1}{2}}) + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.2.2

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

${(x + 1)}^{\frac{1}{2}} x + 3 {(x + 1)}^{\frac{1}{2} + \frac{1}{2}} + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.2.3

Combine the numerators over the common denominator.

${(x + 1)}^{\frac{1}{2}} x + 3 {(x + 1)}^{\frac{1 + 1}{2}} + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.2.4

Add $1$ and $1$ .

${(x + 1)}^{\frac{1}{2}} x + 3 {(x + 1)}^{\frac{2}{2}} + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.2.5

Divide $2$ by $2$ .

${(x + 1)}^{\frac{1}{2}} x + 3 {(x + 1)}^{1} + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.3

Simplify $3 {(x + 1)}^{1}$ .

${(x + 1)}^{\frac{1}{2}} x + 3 (x + 1) + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.4

Apply the distributive property.

${(x + 1)}^{\frac{1}{2}} x + 3 x + 3 \cdot 1 + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.5

Multiply $3$ by $1$ .

${(x + 1)}^{\frac{1}{2}} x + 3 x + 3 + {(x + 1)}^{\frac{1}{2}} \cdot 10 - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.6

Move $10$ to the left of ${(x + 1)}^{\frac{1}{2}}$ .

${(x + 1)}^{\frac{1}{2}} x + 3 x + 3 + 10 \cdot {(x + 1)}^{\frac{1}{2}} - 3 x - 3 (3 {(x + 1)}^{\frac{1}{2}}) - 3 \cdot 10 = 0$

Step 7.2.1.7

Multiply $3$ by $- 3$ .

${(x + 1)}^{\frac{1}{2}} x + 3 x + 3 + 10 {(x + 1)}^{\frac{1}{2}} - 3 x - 9 {(x + 1)}^{\frac{1}{2}} - 3 \cdot 10 = 0$

Step 7.2.1.8

Multiply $- 3$ by $10$ .

${(x + 1)}^{\frac{1}{2}} x + 3 x + 3 + 10 {(x + 1)}^{\frac{1}{2}} - 3 x - 9 {(x + 1)}^{\frac{1}{2}} - 30 = 0$

Step 7.2.2

Simplify by adding terms.

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Step 7.2.2.1

Combine the opposite terms in ${(x + 1)}^{\frac{1}{2}} x + 3 x + 3 + 10 {(x + 1)}^{\frac{1}{2}} - 3 x - 9 {(x + 1)}^{\frac{1}{2}} - 30$ .

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Step 7.2.2.1.1

Subtract $3 x$ from $3 x$ .

${(x + 1)}^{\frac{1}{2}} x + 3 + 10 {(x + 1)}^{\frac{1}{2}} + 0 - 9 {(x + 1)}^{\frac{1}{2}} - 30 = 0$

Step 7.2.2.1.2

Add ${(x + 1)}^{\frac{1}{2}} x + 3 + 10 {(x + 1)}^{\frac{1}{2}}$ and $0$ .

${(x + 1)}^{\frac{1}{2}} x + 3 + 10 {(x + 1)}^{\frac{1}{2}} - 9 {(x + 1)}^{\frac{1}{2}} - 30 = 0$

Step 7.2.2.2

Subtract $30$ from $3$ .

${(x + 1)}^{\frac{1}{2}} x - 27 + 10 {(x + 1)}^{\frac{1}{2}} - 9 {(x + 1)}^{\frac{1}{2}} = 0$

Step 7.2.2.3

Subtract $9 {(x + 1)}^{\frac{1}{2}}$ from $10 {(x + 1)}^{\frac{1}{2}}$ .

${(x + 1)}^{\frac{1}{2}} x - 27 + {(x + 1)}^{\frac{1}{2}} = 0$

Step 7.2.2.4

Reorder factors in ${(x + 1)}^{\frac{1}{2}} x - 27 + {(x + 1)}^{\frac{1}{2}}$ .

$x {(x + 1)}^{\frac{1}{2}} - 27 + {(x + 1)}^{\frac{1}{2}} = 0$

Step 8

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 8$

Step 9