Calculus Examples

$\frac{3}{4} t^{- \frac{1}{4}} - \frac{9}{4} t^{- \frac{3}{4}}$

Step 1

Set $\frac{3}{4} t^{- \frac{1}{4}} - \frac{9}{4} t^{- \frac{3}{4}}$ equal to $0$ .

$\frac{3}{4} t^{- \frac{1}{4}} - \frac{9}{4} t^{- \frac{3}{4}} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$\frac{3}{4} \cdot \frac{1}{t^{\frac{1}{4}}} - \frac{9}{4} t^{- \frac{3}{4}} = 0$

Step 2.1.2

Combine.

$\frac{3 \cdot 1}{4 t^{\frac{1}{4}}} - \frac{9}{4} t^{- \frac{3}{4}} = 0$

Step 2.1.3

Multiply $3$ by $1$ .

$\frac{3}{4 t^{\frac{1}{4}}} - \frac{9}{4} t^{- \frac{3}{4}} = 0$

Step 2.1.4

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

$\frac{3}{4 t^{\frac{1}{4}}} - \frac{9}{4} \cdot \frac{1}{t^{\frac{3}{4}}} = 0$

Step 2.1.5

Multiply $\frac{1}{t^{\frac{3}{4}}}$ by $\frac{9}{4}$ .

$\frac{3}{4 t^{\frac{1}{4}}} - \frac{9}{t^{\frac{3}{4}} \cdot 4} = 0$

Step 2.1.6

Move $4$ to the left of $t^{\frac{3}{4}}$ .

$\frac{3}{4 t^{\frac{1}{4}}} - \frac{9}{4 t^{\frac{3}{4}}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$4 t^{\frac{1}{4}}, 4 t^{\frac{3}{4}}, 1$

Step 2.2.2

Since $4 t^{\frac{1}{4}}, 4 t^{\frac{3}{4}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $4, 4, 1$ then find LCM for the variable part $t^{\frac{1}{4}}, t^{\frac{3}{4}}$ .

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.4

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 2.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.6

The LCM of $4, 4, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2$

Step 2.2.7

Multiply $2$ by $2$ .

$4$

Step 2.2.8

The LCM of $t^{\frac{1}{4}}, t^{\frac{3}{4}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$t^{\frac{3}{4}}$

Step 2.2.9

The LCM for $4 t^{\frac{1}{4}}, 4 t^{\frac{3}{4}}, 1$ is the numeric part $4$ multiplied by the variable part.

$4 t^{\frac{3}{4}}$

Step 2.3

Multiply each term in $\frac{3}{4 t^{\frac{1}{4}}} - \frac{9}{4 t^{\frac{3}{4}}} = 0$ by $4 t^{\frac{3}{4}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{3}{4 t^{\frac{1}{4}}} - \frac{9}{4 t^{\frac{3}{4}}} = 0$ by $4 t^{\frac{3}{4}}$ .

$\frac{3}{4 t^{\frac{1}{4}}} (4 t^{\frac{3}{4}}) - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Rewrite using the commutative property of multiplication.

$4 \frac{3}{4 t^{\frac{1}{4}}} t^{\frac{3}{4}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.2

Cancel the common factor of $4$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$4 \frac{3}{4 t^{\frac{1}{4}}} t^{\frac{3}{4}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.2.2

Rewrite the expression.

$\frac{3}{t^{\frac{1}{4}}} t^{\frac{3}{4}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.3

Cancel the common factor of $t^{\frac{1}{4}}$ .

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Step 2.3.2.1.3.1

Factor $t^{\frac{1}{4}}$ out of $t^{\frac{3}{4}}$ .

$\frac{3}{t^{\frac{1}{4}}} (t^{\frac{1}{4}} t^{\frac{2}{4}}) - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.3.2

Cancel the common factor.

$\frac{3}{t^{\frac{1}{4}}} (t^{\frac{1}{4}} t^{\frac{2}{4}}) - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.3.3

Rewrite the expression.

$3 t^{\frac{2}{4}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.4

Cancel the common factor of $2$ and $4$ .

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Step 2.3.2.1.4.1

Factor $2$ out of $2$ .

$3 t^{\frac{2 (1)}{4}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.4.2

Cancel the common factors.

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Step 2.3.2.1.4.2.1

Factor $2$ out of $4$ .

$3 t^{\frac{2 \cdot 1}{2 \cdot 2}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.4.2.2

Cancel the common factor.

$3 t^{\frac{2 \cdot 1}{2 \cdot 2}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.4.2.3

Rewrite the expression.

$3 t^{\frac{1}{2}} - \frac{9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.5

Cancel the common factor of $4 t^{\frac{3}{4}}$ .

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Step 2.3.2.1.5.1

Move the leading negative in $- \frac{9}{4 t^{\frac{3}{4}}}$ into the numerator.

$3 t^{\frac{1}{2}} + \frac{- 9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.5.2

Cancel the common factor.

$3 t^{\frac{1}{2}} + \frac{- 9}{4 t^{\frac{3}{4}}} (4 t^{\frac{3}{4}}) = 0 (4 t^{\frac{3}{4}})$

Step 2.3.2.1.5.3

Rewrite the expression.

$3 t^{\frac{1}{2}} - 9 = 0 (4 t^{\frac{3}{4}})$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0 (4 t^{\frac{3}{4}})$ .

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Step 2.3.3.1.1

Multiply $4$ by $0$ .

$3 t^{\frac{1}{2}} - 9 = 0 t^{\frac{3}{4}}$

Step 2.3.3.1.2

Multiply $0$ by $t^{\frac{3}{4}}$ .

$3 t^{\frac{1}{2}} - 9 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $9$ to both sides of the equation.

$3 t^{\frac{1}{2}} = 9$

Step 2.4.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(3 t^{\frac{1}{2}})}^{2} = 9^{2}$

Step 2.4.3

Simplify the exponent.

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Step 2.4.3.1

Simplify the left side.

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Step 2.4.3.1.1

Simplify ${(3 t^{\frac{1}{2}})}^{2}$ .

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Step 2.4.3.1.1.1

Apply the product rule to $3 t^{\frac{1}{2}}$ .

$3^{2} {(t^{\frac{1}{2}})}^{2} = 9^{2}$

Step 2.4.3.1.1.2

Raise $3$ to the power of $2$ .

$9 {(t^{\frac{1}{2}})}^{2} = 9^{2}$

Step 2.4.3.1.1.3

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 2.4.3.1.1.3.1

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

$9 t^{\frac{1}{2} \cdot 2} = 9^{2}$

Step 2.4.3.1.1.3.2

Cancel the common factor of $2$ .

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Step 2.4.3.1.1.3.2.1

Cancel the common factor.

$9 t^{\frac{1}{2} \cdot 2} = 9^{2}$

Step 2.4.3.1.1.3.2.2

Rewrite the expression.

$9 t^{1} = 9^{2}$

Step 2.4.3.1.1.4

Simplify.

$9 t = 9^{2}$

Step 2.4.3.2

Simplify the right side.

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Step 2.4.3.2.1

Raise $9$ to the power of $2$ .

$9 t = 81$

Step 2.4.4

Divide each term in $9 t = 81$ by $9$ and simplify.

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Step 2.4.4.1

Divide each term in $9 t = 81$ by $9$ .

$\frac{9 t}{9} = \frac{81}{9}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $9$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{9 t}{9} = \frac{81}{9}$

Step 2.4.4.2.1.2

Divide $t$ by $1$ .

$t = \frac{81}{9}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Divide $81$ by $9$ .

$t = 9$

Step 3