Solve (x-7)^(9/2)-(x-7)^(5/2)

Question

Simplify the expression

x^{4} x - 7 - 28 x^{3} x - 7 + 293 x - 7 \times x^{2} - 1358 x - 7 \times x + 2352 x - 7

Evaluate

(x - 7)^{\frac{9}{2}} - (x - 7)^{\frac{5}{2}}

Simplify the expression

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Evaluate

(x - 7)^{\frac{9}{2}}

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

(x - 7)^{9}

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

(x - 7)^{8 + 1}

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

(x - 7)^{8} (x - 7)

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

(x - 7)^{8} \times x - 7

Reduce the index of the radical and exponent with 2

(x - 7)^{4} x - 7

(x - 7)^{4} x - 7 - (x - 7)^{\frac{5}{2}}

Simplify the expression

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Evaluate

- (x - 7)^{\frac{5}{2}}

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

- (x - 7)^{5}

Simplify the radical expression

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Evaluate

(x - 7)^{5}

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

(x - 7)^{4 + 1}

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

(x - 7)^{4} (x - 7)

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

(x - 7)^{4} \times x - 7

Reduce the index of the radical and exponent with 2

(x - 7)^{2} x - 7

- (x - 7)^{2} x - 7

(x - 7)^{4} x - 7 - (x - 7)^{2} x - 7

Expand the expression

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Evaluate

(x - 7)^{4} x - 7

Expand the expression

(x^{4} - 28 x^{3} + 294 x^{2} - 1372 x + 2401) x - 7

Multiply each term in the parentheses by x - 7

x^{4} x - 7 - 28 x^{3} x - 7 + 294 x^{2} x - 7 - 1372 x x - 7 + 2401 x - 7

x^{4} x - 7 - 28 x^{3} x - 7 + 294 x^{2} x - 7 - 1372 x x - 7 + 2401 x - 7 - (x - 7)^{2} x - 7

Expand the expression

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Evaluate

- (x - 7)^{2} x - 7

Evaluate the power

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Evaluate

- (x - 7)^{2}

Expand the expression

- (x^{2} - 14 x + 49)

Expand the expression

- x^{2} + 14 x - 49

(- x^{2} + 14 x - 49) x - 7

Multiply each term in the parentheses by x - 7

- x^{2} x - 7 + 14 x x - 7 - 49 x - 7

x^{4} x - 7 - 28 x^{3} x - 7 + 294 x^{2} x - 7 - 1372 x x - 7 + 2401 x - 7 - x^{2} x - 7 + 14 x x - 7 - 49 x - 7

Subtract the terms

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Evaluate

294 x^{2} x - 7 - x^{2} x - 7

Rewrite the expression

294 x - 7 \times x^{2} - x - 7 \times x^{2}

Collect like terms by calculating the sum or difference of their coefficients

(294 - 1) x - 7 \times x^{2}

Subtract the numbers

293 x - 7 \times x^{2}

x^{4} x - 7 - 28 x^{3} x - 7 + 293 x - 7 \times x^{2} - 1372 x x - 7 + 2401 x - 7 + 14 x x - 7 - 49 x - 7

Add the terms

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Evaluate

- 1372 x x - 7 + 14 x x - 7

Rewrite the expression

- 1372 x - 7 \times x + 14 x - 7 \times x

Collect like terms by calculating the sum or difference of their coefficients

(- 1372 + 14) x - 7 \times x

Add the numbers

- 1358 x - 7 \times x

x^{4} x - 7 - 28 x^{3} x - 7 + 293 x - 7 \times x^{2} - 1358 x - 7 \times x + 2401 x - 7 - 49 x - 7

Solution

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Evaluate

2401 x - 7 - 49 x - 7

Collect like terms by calculating the sum or difference of their coefficients

(2401 - 49) x - 7

Subtract the numbers

2352 x - 7

x^{4} x - 7 - 28 x^{3} x - 7 + 293 x - 7 \times x^{2} - 1358 x - 7 \times x + 2352 x - 7

Show Solution

Factor the expression

x - 7 \times (x - 7)^{2} (x - 8) (x - 6)

Evaluate

(x - 7)^{\frac{9}{2}} - (x - 7)^{\frac{5}{2}}

Simplify the expression

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Evaluate

(x - 7)^{\frac{9}{2}}

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

(x - 7)^{9}

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

(x - 7)^{8 + 1}

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

(x - 7)^{8} (x - 7)

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

(x - 7)^{8} \times x - 7

Reduce the index of the radical and exponent with 2

(x - 7)^{4} x - 7

(x - 7)^{4} x - 7 - (x - 7)^{\frac{5}{2}}

Simplify the expression

More Steps

Evaluate

- (x - 7)^{\frac{5}{2}}

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

- (x - 7)^{5}

Simplify the radical expression

More Steps

Evaluate

(x - 7)^{5}

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

(x - 7)^{4 + 1}

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

(x - 7)^{4} (x - 7)

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

(x - 7)^{4} \times x - 7

Reduce the index of the radical and exponent with 2

(x - 7)^{2} x - 7

- (x - 7)^{2} x - 7

(x - 7)^{4} x - 7 - (x - 7)^{2} x - 7

Factor out x - 7 \times (x - 7)^{2} from the expression

x - 7 \times (x - 7)^{2} ((x - 7)^{2} - 1)

Solution

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Evaluate

(x - 7)^{2} - 1

Rewrite the expression in exponential form

(x - 7)^{2} - 1^{2}

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

(x - 7 - 1) (x - 7 + 1)

Evaluate

(x - 8) (x - 7 + 1)

Evaluate

(x - 8) (x - 6)

x - 7 \times (x - 7)^{2} (x - 8) (x - 6)

Show Solution

Find the roots

x_{1} = 7, x_{2} = 8

Evaluate

(x - 7)^{\frac{9}{2}} - (x - 7)^{\frac{5}{2}}

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

(x - 7)^{\frac{9}{2}} - (x - 7)^{\frac{5}{2}} = 0

Find the domain

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Evaluate

x - 7 \geq 0

Move the constant to the right side

x \geq 0 + 7

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

x \geq 7

(x - 7)^{\frac{9}{2}} - (x - 7)^{\frac{5}{2}} = 0, x \geq 7

Calculate

(x - 7)^{\frac{9}{2}} - (x - 7)^{\frac{5}{2}} = 0

Factor the expression

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Evaluate

(x - 7)^{\frac{9}{2}} - (x - 7)^{\frac{5}{2}}

Factor out (x - 7)^{\frac{5}{2}} from the expression

(x - 7)^{\frac{5}{2}} ((x - 7)^{2} - 1)

Factor the expression

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Evaluate

(x - 7)^{2} - 1

Rewrite the expression in exponential form

(x - 7)^{2} - 1^{2}

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

(x - 7 - 1) (x - 7 + 1)

Evaluate

(x - 8) (x - 7 + 1)

Evaluate

(x - 8) (x - 6)

(x - 7)^{\frac{5}{2}} (x - 8) (x - 6)

(x - 7)^{\frac{5}{2}} (x - 8) (x - 6) = 0

Separate the equation into 3 possible cases

(x - 7)^{\frac{5}{2}} = 0 x - 8 = 0 x - 6 = 0

Solve the equation

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Evaluate

(x - 7)^{\frac{5}{2}} = 0

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

x - 7 = 0

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

x = 0 + 7

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

x = 7

x = 7 x - 8 = 0 x - 6 = 0

Solve the equation

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Evaluate

x - 8 = 0

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

x = 0 + 8

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

x = 8

x = 7 x = 8 x - 6 = 0

Solve the equation

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Evaluate

x - 6 = 0

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

x = 0 + 6

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

x = 6

x = 7 x = 8 x = 6

Check if the solution is in the defined range

x = 7 x = 8 x = 6, x \geq 7

Find the intersection of the solution and the defined range

x = 7 x = 8

Solution

x_{1} = 7, x_{2} = 8

Show Solution