Finite Math Examples

$x y^{2} = 10^{11}$ , $\frac{x^{3}}{y} = 10^{19}$

Step 1

Divide each term in $x y^{2} = 10^{11}$ by $y^{2}$ and simplify.

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Step 1.1

Divide each term in $x y^{2} = 10^{11}$ by $y^{2}$ .

$\frac{x y^{2}}{y^{2}} = \frac{10^{11}}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $y^{2}$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{x y^{2}}{y^{2}} = \frac{10^{11}}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

Step 1.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{11}}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

$x = \frac{10^{11}}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

$x = \frac{10^{11}}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Raise $10$ to the power of $11$ .

$x = \frac{100000000000}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{x^{3}}{y} = 10^{19}$

Step 2

Replace all occurrences of $x$ with $\frac{100000000000}{y^{2}}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $\frac{x^{3}}{y} = 10^{19}$ with $\frac{100000000000}{y^{2}}$ .

$\frac{{(\frac{100000000000}{y^{2}})}^{3}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2

Simplify $\frac{{(\frac{100000000000}{y^{2}})}^{3}}{y} = 10^{19}$ .

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $\frac{{(\frac{100000000000}{y^{2}})}^{3}}{y}$ .

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Step 2.2.1.1.1

Simplify the numerator.

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Step 2.2.1.1.1.1

Apply the product rule to $\frac{100000000000}{y^{2}}$ .

$\frac{\frac{100000000000^{3}}{{(y^{2})}^{3}}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.1.2

Multiply the exponents in ${(y^{2})}^{3}$ .

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Step 2.2.1.1.1.2.1

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

$\frac{\frac{100000000000^{3}}{y^{2 \cdot 3}}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.1.2.2

Multiply $2$ by $3$ .

$\frac{\frac{100000000000^{3}}{y^{6}}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{\frac{100000000000^{3}}{y^{6}}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{\frac{100000000000^{3}}{y^{6}}}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{100000000000^{3}}{y^{6}} \cdot \frac{1}{y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.3

Combine.

$\frac{100000000000^{3} \cdot 1}{y^{6} y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.4

Multiply $y^{6}$ by $y$ by adding the exponents.

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Step 2.2.1.1.4.1

Multiply $y^{6}$ by $y$ .

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Step 2.2.1.1.4.1.1

Raise $y$ to the power of $1$ .

$\frac{100000000000^{3} \cdot 1}{y^{6} y} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.4.1.2

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

$\frac{100000000000^{3} \cdot 1}{y^{6 + 1}} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{100000000000^{3} \cdot 1}{y^{6 + 1}} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.4.2

Add $6$ and $1$ .

$\frac{100000000000^{3} \cdot 1}{y^{7}} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{100000000000^{3} \cdot 1}{y^{7}} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.1.1.5

Multiply $100000000000^{3}$ by $1$ .

$\frac{100000000000^{3}}{y^{7}} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{100000000000^{3}}{y^{7}} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

$\frac{100000000000^{3}}{y^{7}} = 10^{19}$

$x = \frac{100000000000}{y^{2}}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Raise $10$ to the power of $19$ .

$\frac{100000000000^{3}}{y^{7}} = 10000000000000000000$

$x = \frac{100000000000}{y^{2}}$

$\frac{100000000000^{3}}{y^{7}} = 10000000000000000000$

$x = \frac{100000000000}{y^{2}}$

$\frac{100000000000^{3}}{y^{7}} = 10000000000000000000$

$x = \frac{100000000000}{y^{2}}$

$\frac{100000000000^{3}}{y^{7}} = 10000000000000000000$

$x = \frac{100000000000}{y^{2}}$

Step 3

Solve for $y$ in $\frac{100000000000^{3}}{y^{7}} = 10000000000000000000$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{7}, 1$

$x = \frac{100000000000}{y^{2}}$

Step 3.1.2

The LCM of one and any expression is the expression.

$y^{7}$

$x = \frac{100000000000}{y^{2}}$

$y^{7}$

$x = \frac{100000000000}{y^{2}}$

Step 3.2

Multiply each term in $\frac{100000000000^{3}}{y^{7}} = 10000000000000000000$ by $y^{7}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{100000000000^{3}}{y^{7}} = 10000000000000000000$ by $y^{7}$ .

$\frac{100000000000^{3}}{y^{7}} \cdot y^{7} = 10000000000000000000 y^{7}$

$x = \frac{100000000000}{y^{2}}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $y^{7}$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{100000000000^{3}}{y^{7}} \cdot y^{7} = 10000000000000000000 y^{7}$

$x = \frac{100000000000}{y^{2}}$

Step 3.2.2.1.2

Rewrite the expression.

$100000000000^{3} = 10000000000000000000 y^{7}$

$x = \frac{100000000000}{y^{2}}$

$100000000000^{3} = 10000000000000000000 y^{7}$

$x = \frac{100000000000}{y^{2}}$

$100000000000^{3} = 10000000000000000000 y^{7}$

$x = \frac{100000000000}{y^{2}}$

$100000000000^{3} = 10000000000000000000 y^{7}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $10000000000000000000 y^{7} = 100000000000^{3}$ .

$10000000000000000000 y^{7} = 100000000000^{3}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.2

Divide each term in $10000000000000000000 y^{7} = 100000000000^{3}$ by $10000000000000000000$ and simplify.

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Step 3.3.2.1

Divide each term in $10000000000000000000 y^{7} = 100000000000^{3}$ by $10000000000000000000$ .

$\frac{10000000000000000000 y^{7}}{10000000000000000000} = \frac{100000000000^{3}}{10000000000000000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $10000000000000000000$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{10000000000000000000 y^{7}}{10000000000000000000} = \frac{100000000000^{3}}{10000000000000000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.2.2.1.2

Divide $y^{7}$ by $1$ .

$y^{7} = \frac{100000000000^{3}}{10000000000000000000}$

$x = \frac{100000000000}{y^{2}}$

$y^{7} = \frac{100000000000^{3}}{10000000000000000000}$

$x = \frac{100000000000}{y^{2}}$

$y^{7} = \frac{100000000000^{3}}{10000000000000000000}$

$x = \frac{100000000000}{y^{2}}$

$y^{7} = \frac{100000000000^{3}}{10000000000000000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[7]{\frac{100000000000^{3}}{10000000000000000000}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4

Simplify $\sqrt[7]{\frac{100000000000^{3}}{10000000000000000000}}$ .

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Step 3.3.4.1

Rewrite $\sqrt[7]{\frac{100000000000^{3}}{10000000000000000000}}$ as $\frac{\sqrt[7]{100000000000^{3}}}{\sqrt[7]{10000000000000000000}}$ .

$y = \frac{\sqrt[7]{100000000000^{3}}}{\sqrt[7]{10000000000000000000}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.2

Simplify the denominator.

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Step 3.3.4.2.1

Rewrite $10000000000000000000$ as $100^{7} \cdot 100000$ .

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Step 3.3.4.2.1.1

Factor $100000000000000$ out of $10000000000000000000$ .

$y = \frac{\sqrt[7]{100000000000^{3}}}{\sqrt[7]{100000000000000 (100000)}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.2.1.2

Rewrite $100000000000000$ as $100^{7}$ .

$y = \frac{\sqrt[7]{100000000000^{3}}}{\sqrt[7]{100^{7} \cdot 100000}}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{100000000000^{3}}}{\sqrt[7]{100^{7} \cdot 100000}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.2.2

Pull terms out from under the radical.

$y = \frac{\sqrt[7]{100000000000^{3}}}{100 \sqrt[7]{100000}}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{100000000000^{3}}}{100 \sqrt[7]{100000}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.3

Multiply $\frac{\sqrt[7]{100000000000^{3}}}{100 \sqrt[7]{100000}}$ by $\frac{{\sqrt[7]{100000}}^{6}}{{\sqrt[7]{100000}}^{6}}$ .

$y = \frac{\sqrt[7]{100000000000^{3}}}{100 \sqrt[7]{100000}} \cdot \frac{{\sqrt[7]{100000}}^{6}}{{\sqrt[7]{100000}}^{6}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4

Combine and simplify the denominator.

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Step 3.3.4.4.1

Multiply $\frac{\sqrt[7]{100000000000^{3}}}{100 \sqrt[7]{100000}}$ by $\frac{{\sqrt[7]{100000}}^{6}}{{\sqrt[7]{100000}}^{6}}$ .

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \sqrt[7]{100000} {\sqrt[7]{100000}}^{6}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.2

Move $\sqrt[7]{100000}$ .

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 (\sqrt[7]{100000} {\sqrt[7]{100000}}^{6})}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.3

Raise $\sqrt[7]{100000}$ to the power of $1$ .

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 (\sqrt[7]{100000} {\sqrt[7]{100000}}^{6})}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.4

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

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 {\sqrt[7]{100000}}^{1 + 6}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.5

Add $1$ and $6$ .

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 {\sqrt[7]{100000}}^{7}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.6

Rewrite ${\sqrt[7]{100000}}^{7}$ as $100000$ .

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Step 3.3.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{100000}$ as $100000^{\frac{1}{7}}$ .

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 {(100000^{\frac{1}{7}})}^{7}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.6.2

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

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000^{\frac{1}{7} \cdot 7}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.6.3

Combine $\frac{1}{7}$ and $7$ .

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000^{\frac{7}{7}}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.6.4

Cancel the common factor of $7$ .

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Step 3.3.4.4.6.4.1

Cancel the common factor.

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000^{\frac{7}{7}}}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.6.4.2

Rewrite the expression.

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.4.6.5

Evaluate the exponent.

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{100000000000^{3}} {\sqrt[7]{100000}}^{6}}{100 \cdot 100000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.5

Rewrite ${\sqrt[7]{100000}}^{6}$ as $\sqrt[7]{100000^{6}}$ .

$y = \frac{\sqrt[7]{100000000000^{3}} \sqrt[7]{100000^{6}}}{100 \cdot 100000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.6

Multiply $100$ by $100000$ .

$y = \frac{\sqrt[7]{100000000000^{3}} \sqrt[7]{100000^{6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7

Simplify the numerator.

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Step 3.3.4.7.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[7]{100000000000^{3} \cdot 100000^{6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.2

Rewrite $100000000000$ as $10^{11}$ .

$y = \frac{\sqrt[7]{{(10^{11})}^{3} \cdot 100000^{6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.3

Multiply the exponents in ${(10^{11})}^{3}$ .

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Step 3.3.4.7.3.1

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

$y = \frac{\sqrt[7]{10^{11 \cdot 3} \cdot 100000^{6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.3.2

Multiply $11$ by $3$ .

$y = \frac{\sqrt[7]{10^{33} \cdot 100000^{6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{10^{33} \cdot 100000^{6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.4

Rewrite $100000$ as $10^{5}$ .

$y = \frac{\sqrt[7]{10^{33} \cdot {(10^{5})}^{6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.5

Multiply the exponents in ${(10^{5})}^{6}$ .

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Step 3.3.4.7.5.1

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

$y = \frac{\sqrt[7]{10^{33} \cdot 10^{5 \cdot 6}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.5.2

Multiply $5$ by $6$ .

$y = \frac{\sqrt[7]{10^{33} \cdot 10^{30}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{10^{33} \cdot 10^{30}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.6

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

$y = \frac{\sqrt[7]{10^{33 + 30}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.7.7

Add $33$ and $30$ .

$y = \frac{\sqrt[7]{10^{63}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{\sqrt[7]{10^{63}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.8

Simplify the numerator.

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Step 3.3.4.8.1

Rewrite $10^{63}$ as ${(10^{9})}^{7}$ .

$y = \frac{\sqrt[7]{{(10^{9})}^{7}}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.8.2

Pull terms out from under the radical, assuming real numbers.

$y = \frac{10^{9}}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.8.3

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

$y = \frac{1000000000}{10000000}$

$x = \frac{100000000000}{y^{2}}$

$y = \frac{1000000000}{10000000}$

$x = \frac{100000000000}{y^{2}}$

Step 3.3.4.9

Divide $1000000000$ by $10000000$ .

$y = 100$

$x = \frac{100000000000}{y^{2}}$

$y = 100$

$x = \frac{100000000000}{y^{2}}$

$y = 100$

$x = \frac{100000000000}{y^{2}}$

$y = 100$

$x = \frac{100000000000}{y^{2}}$

Step 4

Replace all occurrences of $y$ with $100$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{100000000000}{y^{2}}$ with $100$ .

$x = \frac{100000000000}{{(100)}^{2}}$

$y = 100$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{100000000000}{{(100)}^{2}}$ .

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Step 4.2.1.1

Raise $100$ to the power of $2$ .

$x = \frac{100000000000}{10000}$

$y = 100$

Step 4.2.1.2

Divide $100000000000$ by $10000$ .

$x = 10000000$

$y = 100$

$x = 10000000$

$y = 100$

$x = 10000000$

$y = 100$

$x = 10000000$

$y = 100$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(10000000, 100)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(10000000, 100)$

Equation Form:

$x = 10000000, y = 100$

Step 7