Pre-Algebra Examples

$4.5 x \cdot 10^{9} x^{8} \cdot \frac{1}{2}$

Step 1

Write $4.5 x \cdot 10^{9} x^{8} \cdot \frac{1}{2}$ as a function.

$f (x) = 4.5 x \cdot 10^{9} x^{8} \cdot \frac{1}{2}$

Step 2

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

$4.5, 10^{9}, \frac{1}{2}$

Step 3

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 $1$ .

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Step 3.1

Arrange the terms in ascending order.

$b_{1} = | \frac{1}{2} |, | 4.5 |, | 10^{9} |$

Step 3.2

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

$b_{1} = | 10^{9} |$

Step 3.3

Simplify each term.

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Step 3.3.1

Raise $10$ to the power of $9$ .

$b_{1} = | 1000000000 | + 1$

Step 3.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1000000000$ is $1000000000$ .

$b_{1} = 1000000000 + 1$

Step 3.4

Add $1000000000$ and $1$ .

$b_{1} = 1000000001$

Step 4

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 $1$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4.5$ is $4.5$ .

$b_{2} = 4.5 + | 10^{9} | + | \frac{1}{2} |$

Step 4.1.2

Raise $10$ to the power of $9$ .

$b_{2} = 4.5 + | 1000000000 | + | \frac{1}{2} |$

Step 4.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1000000000$ is $1000000000$ .

$b_{2} = 4.5 + 1000000000 + | \frac{1}{2} |$

Step 4.1.4

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$b_{2} = 4.5 + 1000000000 + \frac{1}{2}$

Step 4.2

Find the common denominator.

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Step 4.2.1

Write $4.5$ as a fraction with denominator $1$ .

$b_{2} = \frac{4.5}{1} + 1000000000 + \frac{1}{2}$

Step 4.2.2

Multiply $\frac{4.5}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{4.5}{1} \cdot \frac{2}{2} + 1000000000 + \frac{1}{2}$

Step 4.2.3

Multiply $\frac{4.5}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{4.5 \cdot 2}{2} + 1000000000 + \frac{1}{2}$

Step 4.2.4

Write $1000000000$ as a fraction with denominator $1$ .

$b_{2} = \frac{4.5 \cdot 2}{2} + \frac{1000000000}{1} + \frac{1}{2}$

Step 4.2.5

Multiply $\frac{1000000000}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{4.5 \cdot 2}{2} + \frac{1000000000}{1} \cdot \frac{2}{2} + \frac{1}{2}$

Step 4.2.6

Multiply $\frac{1000000000}{1}$ by $\frac{2}{2}$ .

$b_{2} = \frac{4.5 \cdot 2}{2} + \frac{1000000000 \cdot 2}{2} + \frac{1}{2}$

Step 4.3

Combine the numerators over the common denominator.

$b_{2} = \frac{4.5 \cdot 2 + 1000000000 \cdot 2 + 1}{2}$

Step 4.4

Simplify each term.

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Step 4.4.1

Multiply $4.5$ by $2$ .

$b_{2} = \frac{9 + 1000000000 \cdot 2 + 1}{2}$

Step 4.4.2

Multiply $1000000000$ by $2$ .

$b_{2} = \frac{9 + 2000000000 + 1}{2}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Add $9$ and $2000000000$ .

$b_{2} = \frac{2000000009 + 1}{2}$

Step 4.5.2

Add $2000000009$ and $1$ .

$b_{2} = \frac{2000000010}{2}$

Step 4.5.3

Divide $2000000010$ by $2$ .

$b_{2} = 1000000005$

Step 4.6

Arrange the terms in ascending order.

$b_{2} = 1, 1000000005$

Step 4.7

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

$b_{2} = 1000000005$

Step 5

Take the smaller bound option between $b_{1} = 1000000001$ and $b_{2} = 1000000005$ .

Smaller Bound: $1000000001$

Step 6

Every real root on $f (x) = 4.5 x \cdot (10^{9} x^{8}) \cdot \frac{1}{2}$ lies between $- 1000000001$ and $1000000001$ .

$- 1000000001$ and $1000000001$