Solve (x-3)^(5/2)-(x-3)^(1/2)

Question

Simplify the expression

x^{2} x - 3 - 6 x x - 3 + 8 x - 3

Evaluate

(x - 3)^{\frac{5}{2}} - (x - 3)^{\frac{1}{2}}

Simplify the expression

More Steps

Evaluate

(x - 3)^{\frac{5}{2}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

(x - 3)^{5}

Rewrite the exponent as a sum where one of the addends is a multiple of the index

(x - 3)^{4 + 1}

Use a^{m + n} = a^{m} \times a^{n} to expand the expression

(x - 3)^{4} (x - 3)

The root of a product is equal to the product of the roots of each factor

(x - 3)^{4} \times x - 3

Reduce the index of the radical and exponent with 2

(x - 3)^{2} x - 3

(x - 3)^{2} x - 3 - (x - 3)^{\frac{1}{2}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

(x - 3)^{2} x - 3 - x - 3

Expand the expression

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Evaluate

(x - 3)^{2} x - 3

Expand the expression

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Evaluate

(x - 3)^{2}

Use (a - b)^{2} = a^{2} - 2 ab + b^{2} to expand the expression

x^{2} - 2 x \times 3 + 3^{2}

Calculate

x^{2} - 6 x + 9

(x^{2} - 6 x + 9) x - 3

Multiply each term in the parentheses by x - 3

x^{2} x - 3 - 6 x x - 3 + 9 x - 3

x^{2} x - 3 - 6 x x - 3 + 9 x - 3 - x - 3

Solution

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Evaluate

9 x - 3 - x - 3

Factor the expression

(9 - 1) x - 3

Subtract the terms

8 x - 3

x^{2} x - 3 - 6 x x - 3 + 8 x - 3

Show Solution

Factor the expression

x - 3 \times (x - 4) (x - 2)

Evaluate

(x - 3)^{\frac{5}{2}} - (x - 3)^{\frac{1}{2}}

Simplify the expression

More Steps

Evaluate

(x - 3)^{\frac{5}{2}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

(x - 3)^{5}

Rewrite the exponent as a sum where one of the addends is a multiple of the index

(x - 3)^{4 + 1}

Use a^{m + n} = a^{m} \times a^{n} to expand the expression

(x - 3)^{4} (x - 3)

The root of a product is equal to the product of the roots of each factor

(x - 3)^{4} \times x - 3

Reduce the index of the radical and exponent with 2

(x - 3)^{2} x - 3

(x - 3)^{2} x - 3 - (x - 3)^{\frac{1}{2}}

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

(x - 3)^{2} x - 3 - x - 3

Factor out x - 3 from the expression

x - 3 \times ((x - 3)^{2} - 1)

Solution

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Evaluate

(x - 3)^{2} - 1

Rewrite the expression in exponential form

(x - 3)^{2} - 1^{2}

Use a^{2} - b^{2} = (a - b) (a + b) to factor the expression

(x - 3 - 1) (x - 3 + 1)

Evaluate

(x - 4) (x - 3 + 1)

Evaluate

(x - 4) (x - 2)

x - 3 \times (x - 4) (x - 2)

Show Solution

Find the roots

x_{1} = 3, x_{2} = 4

Evaluate

(x - 3)^{\frac{5}{2}} - (x - 3)^{\frac{1}{2}}

To find the roots of the expression,set the expression equal to 0

(x - 3)^{\frac{5}{2}} - (x - 3)^{\frac{1}{2}} = 0

Find the domain

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Evaluate

x - 3 \geq 0

Move the constant to the right side

x \geq 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

x \geq 3

(x - 3)^{\frac{5}{2}} - (x - 3)^{\frac{1}{2}} = 0, x \geq 3

Calculate

(x - 3)^{\frac{5}{2}} - (x - 3)^{\frac{1}{2}} = 0

Factor the expression

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Evaluate

(x - 3)^{\frac{5}{2}} - (x - 3)^{\frac{1}{2}}

Factor out (x - 3)^{\frac{1}{2}} from the expression

(x - 3)^{\frac{1}{2}} ((x - 3)^{2} - 1)

Factor the expression

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Evaluate

(x - 3)^{2} - 1

Rewrite the expression in exponential form

(x - 3)^{2} - 1^{2}

Use a^{2} - b^{2} = (a - b) (a + b) to factor the expression

(x - 3 - 1) (x - 3 + 1)

Evaluate

(x - 4) (x - 3 + 1)

Evaluate

(x - 4) (x - 2)

(x - 3)^{\frac{1}{2}} (x - 4) (x - 2)

(x - 3)^{\frac{1}{2}} (x - 4) (x - 2) = 0

Separate the equation into 3 possible cases

(x - 3)^{\frac{1}{2}} = 0 x - 4 = 0 x - 2 = 0

Solve the equation

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Evaluate

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

The only way a root could be 0 is when the radicand equals 0

x - 3 = 0

Move the constant to the right-hand side and change its sign

x = 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

x = 3

x = 3 x - 4 = 0 x - 2 = 0

Solve the equation

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Evaluate

x - 4 = 0

Move the constant to the right-hand side and change its sign

x = 0 + 4

Removing 0 doesn't change the value,so remove it from the expression

x = 4

x = 3 x = 4 x - 2 = 0

Solve the equation

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Evaluate

x - 2 = 0

Move the constant to the right-hand side and change its sign

x = 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

x = 2

x = 3 x = 4 x = 2

Check if the solution is in the defined range

x = 3 x = 4 x = 2, x \geq 3

Find the intersection of the solution and the defined range

x = 3 x = 4

Solution

x_{1} = 3, x_{2} = 4

Show Solution