Pre-Algebra Examples

$12 \sqrt[4]{x} - 17 \sqrt[8]{x} + 6$

Step 1

Write $12 \sqrt[4]{x} - 17 \sqrt[8]{x} + 6$ as a function.

$f (x) = 12 \sqrt[4]{x} - 17 \sqrt[8]{x} + 6$

Step 2

Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.

Largest Degree: $- 1$

Leading Coefficient: $12$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

$f (x) = \frac{12 \sqrt[4]{x}}{12} + \frac{- 17 \sqrt[8]{x}}{12} + \frac{6}{12}$

Step 3.1.2

Divide $\sqrt[4]{x}$ by $1$ .

$f (x) = \sqrt[4]{x} + \frac{- 17 \sqrt[8]{x}}{12} + \frac{6}{12}$

Step 3.2

Move the negative in front of the fraction.

$f (x) = \sqrt[4]{x} - \frac{17 \sqrt[8]{x}}{12} + \frac{6}{12}$

Step 3.3

Cancel the common factor of $6$ and $12$ .

Tap for more steps...

Step 3.3.1

Factor $6$ out of $6$ .

$f (x) = \sqrt[4]{x} - \frac{17 \sqrt[8]{x}}{12} + \frac{6 (1)}{12}$

Step 3.3.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1

Factor $6$ out of $12$ .

$f (x) = \sqrt[4]{x} - \frac{17 \sqrt[8]{x}}{12} + \frac{6 \cdot 1}{6 \cdot 2}$

Step 3.3.2.2

Cancel the common factor.

$f (x) = \sqrt[4]{x} - \frac{17 \sqrt[8]{x}}{12} + \frac{6 \cdot 1}{6 \cdot 2}$

Step 3.3.2.3

Rewrite the expression.

$f (x) = \sqrt[4]{x} - \frac{17 \sqrt[8]{x}}{12} + \frac{1}{2}$

Step 4

Check the leading coefficient of the function. This number is the coefficient of the expression with the largest degree.

Largest Degree: $- 1$

Leading Coefficient: $- 17$

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Move the negative in front of the fraction.

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{- \frac{17 \sqrt[8]{x}}{12}}{- 17} + \frac{\frac{1}{2}}{- 17}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$f (x) = - \frac{\sqrt[4]{x}}{17} - \frac{17 \sqrt[8]{x}}{12} \cdot \frac{1}{- 17} + \frac{\frac{1}{2}}{- 17}$

Step 5.3

Cancel the common factor of $17$ .

Tap for more steps...

Step 5.3.1

Move the leading negative in $- \frac{17 \sqrt[8]{x}}{12}$ into the numerator.

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{- 17 \sqrt[8]{x}}{12} \cdot \frac{1}{- 17} + \frac{\frac{1}{2}}{- 17}$

Step 5.3.2

Factor $17$ out of $- 17 \sqrt[8]{x}$ .

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{17 (- \sqrt[8]{x})}{12} \cdot \frac{1}{- 17} + \frac{\frac{1}{2}}{- 17}$

Step 5.3.3

Factor $17$ out of $- 17$ .

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{17 (- \sqrt[8]{x})}{12} \cdot \frac{1}{17 (- 1)} + \frac{\frac{1}{2}}{- 17}$

Step 5.3.4

Cancel the common factor.

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{17 (- \sqrt[8]{x})}{12} \cdot \frac{1}{17 \cdot - 1} + \frac{\frac{1}{2}}{- 17}$

Step 5.3.5

Rewrite the expression.

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{- \sqrt[8]{x}}{12} \cdot \frac{1}{- 1} + \frac{\frac{1}{2}}{- 17}$

Step 5.4

Multiply $\frac{- \sqrt[8]{x}}{12}$ by $\frac{1}{- 1}$ .

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{- \sqrt[8]{x}}{12 \cdot - 1} + \frac{\frac{1}{2}}{- 17}$

Step 5.5

Multiply $12$ by $- 1$ .

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{- \sqrt[8]{x}}{- 12} + \frac{\frac{1}{2}}{- 17}$

Step 5.6

Dividing two negative values results in a positive value.

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{\sqrt[8]{x}}{12} + \frac{\frac{1}{2}}{- 17}$

Step 5.7

Multiply the numerator by the reciprocal of the denominator.

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{\sqrt[8]{x}}{12} + \frac{1}{2} \cdot \frac{1}{- 17}$

Step 5.8

Move the negative in front of the fraction.

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{\sqrt[8]{x}}{12} + \frac{1}{2} \cdot (- \frac{1}{17})$

Step 5.9

Multiply $\frac{1}{2} (- \frac{1}{17})$ .

Tap for more steps...

Step 5.9.1

Multiply $\frac{1}{2}$ by $\frac{1}{17}$ .

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{\sqrt[8]{x}}{12} - \frac{1}{2 \cdot 17}$

Step 5.9.2

Multiply $2$ by $17$ .

$f (x) = - \frac{\sqrt[4]{x}}{17} + \frac{\sqrt[8]{x}}{12} - \frac{1}{34}$

Step 6

Create a list of the coefficients of the function except the leading coefficient of $1$ .

$- \frac{1}{34}$

Step 7

There will be two bound options, $b_{1}$ and $b_{2}$ , the smaller of which is the answer. To calculate the first bound option, find the absolute value of the largest coefficient from the list of coefficients. Then add $1$ .

Tap for more steps...

Step 7.1

Arrange the terms in ascending order.

$b_{1} = | - \frac{1}{34} |$

Step 7.2

$- \frac{1}{34}$ is approximately $- 0.02941176$ which is negative so negate $- \frac{1}{34}$ and remove the absolute value

$b_{1} = \frac{1}{34} + 1$

Step 7.3

Write $1$ as a fraction with a common denominator.

$b_{1} = \frac{1}{34} + \frac{34}{34}$

Step 7.4

Combine the numerators over the common denominator.

$b_{1} = \frac{1 + 34}{34}$

Step 7.5

Add $1$ and $34$ .

$b_{1} = \frac{35}{34}$

Step 8

To calculate the second bound option, sum the absolute values of the coefficients from the list of coefficients. If the sum is greater than $1$ , use that number. If not, use $1$ .

Tap for more steps...

Step 8.1

$- \frac{1}{34}$ is approximately $- 0.02941176$ which is negative so negate $- \frac{1}{34}$ and remove the absolute value

$b_{2} = \frac{1}{34}$

Step 8.2

Arrange the terms in ascending order.

$b_{2} = \frac{1}{34}, 1$

Step 8.3

The maximum value is the largest value in the arranged data set.

$b_{2} = 1$

Step 9

Take the smaller bound option between $b_{1} = \frac{35}{34}$ and $b_{2} = 1$ .

Smaller Bound: $1$

Step 10

Every real root on $f (x) = 12 \sqrt[4]{x} - 17 \sqrt[8]{x} + 6$ lies between $- 1$ and $1$ .

$- 1$ and $1$