Calculus Examples

$7 - 2 t^{- 2} = 0$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$7 - 2 \frac{1}{t^{2}} = 0$

Step 1.2

Combine $- 2$ and $\frac{1}{t^{2}}$ .

$7 + \frac{- 2}{t^{2}} = 0$

Step 1.3

Move the negative in front of the fraction.

$7 - \frac{2}{t^{2}} = 0$

Step 2

Subtract $7$ from both sides of the equation.

$- \frac{2}{t^{2}} = - 7$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$t^{2}, 1$

Step 3.2

The LCM of one and any expression is the expression.

$t^{2}$

Step 4

Multiply each term in $- \frac{2}{t^{2}} = - 7$ by $t^{2}$ to eliminate the fractions.

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Step 4.1

Multiply each term in $- \frac{2}{t^{2}} = - 7$ by $t^{2}$ .

$- \frac{2}{t^{2}} t^{2} = - 7 t^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Move the leading negative in $- \frac{2}{t^{2}}$ into the numerator.

$\frac{- 2}{t^{2}} t^{2} = - 7 t^{2}$

Step 4.2.1.2

Cancel the common factor.

$\frac{- 2}{t^{2}} t^{2} = - 7 t^{2}$

Step 4.2.1.3

Rewrite the expression.

$- 2 = - 7 t^{2}$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $- 7 t^{2} = - 2$ .

$- 7 t^{2} = - 2$

Step 5.2

Divide each term in $- 7 t^{2} = - 2$ by $- 7$ and simplify.

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Step 5.2.1

Divide each term in $- 7 t^{2} = - 2$ by $- 7$ .

$\frac{- 7 t^{2}}{- 7} = \frac{- 2}{- 7}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 7$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 7 t^{2}}{- 7} = \frac{- 2}{- 7}$

Step 5.2.2.1.2

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{- 2}{- 7}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Dividing two negative values results in a positive value.

$t^{2} = \frac{2}{7}$

Step 5.3

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

$t = \pm \sqrt{\frac{2}{7}}$

Step 5.4

Simplify $\pm \sqrt{\frac{2}{7}}$ .

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Step 5.4.1

Rewrite $\sqrt{\frac{2}{7}}$ as $\frac{\sqrt{2}}{\sqrt{7}}$ .

$t = \pm \frac{\sqrt{2}}{\sqrt{7}}$

Step 5.4.2

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$t = \pm \frac{\sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 5.4.3

Combine and simplify the denominator.

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Step 5.4.3.1

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$t = \pm \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 5.4.3.2

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

$t = \pm \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 5.4.3.3

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

$t = \pm \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 5.4.3.4

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

$t = \pm \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 5.4.3.5

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 5.4.3.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$t = \pm \frac{\sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 5.4.3.6.2

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

$t = \pm \frac{\sqrt{2} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 5.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$t = \pm \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.4.3.6.4

Cancel the common factor of $2$ .

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Step 5.4.3.6.4.1

Cancel the common factor.

$t = \pm \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.4.3.6.4.2

Rewrite the expression.

$t = \pm \frac{\sqrt{2} \sqrt{7}}{7^{1}}$

Step 5.4.3.6.5

Evaluate the exponent.

$t = \pm \frac{\sqrt{2} \sqrt{7}}{7}$

Step 5.4.4

Simplify the numerator.

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Step 5.4.4.1

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{2 \cdot 7}}{7}$

Step 5.4.4.2

Multiply $2$ by $7$ .

$t = \pm \frac{\sqrt{14}}{7}$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{\sqrt{14}}{7}$

Step 5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{\sqrt{14}}{7}$

Step 5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{\sqrt{14}}{7}, - \frac{\sqrt{14}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$t = \frac{\sqrt{14}}{7}, - \frac{\sqrt{14}}{7}$

Decimal Form:

$t = 0.53452248 \dots, - 0.53452248 \dots$