Calculus Examples

$lim_{x \to 3} (\frac{7}{12 \sqrt[12]{x^{5}}}) - 8 \sqrt[3]{x} + \frac{1}{\sqrt[3]{x^{2}}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$lim_{x \to 3} \frac{7}{12 \sqrt[12]{x^{5}}} - lim_{x \to 3} 8 \sqrt[3]{x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 2

Move the term $\frac{7}{12}$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{12} lim_{x \to 3} \frac{1}{\sqrt[12]{x^{5}}} - lim_{x \to 3} 8 \sqrt[3]{x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{7}{12} \cdot \frac{lim_{x \to 3} 1}{lim_{x \to 3} \sqrt[12]{x^{5}}} - lim_{x \to 3} 8 \sqrt[3]{x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\frac{7}{12} \cdot \frac{1}{lim_{x \to 3} \sqrt[12]{x^{5}}} - lim_{x \to 3} 8 \sqrt[3]{x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 5

Move the limit under the radical sign.

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{lim_{x \to 3} x^{5}}} - lim_{x \to 3} 8 \sqrt[3]{x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 6

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{{(lim_{x \to 3} x)}^{5}}} - lim_{x \to 3} 8 \sqrt[3]{x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 7

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{{(lim_{x \to 3} x)}^{5}}} - 8 lim_{x \to 3} \sqrt[3]{x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 8

Move the limit under the radical sign.

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{{(lim_{x \to 3} x)}^{5}}} - 8 \sqrt[3]{lim_{x \to 3} x} + lim_{x \to 3} \frac{1}{\sqrt[3]{x^{2}}}$

Step 9

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{{(lim_{x \to 3} x)}^{5}}} - 8 \sqrt[3]{lim_{x \to 3} x} + \frac{lim_{x \to 3} 1}{lim_{x \to 3} \sqrt[3]{x^{2}}}$

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{{(lim_{x \to 3} x)}^{5}}} - 8 \sqrt[3]{lim_{x \to 3} x} + \frac{1}{lim_{x \to 3} \sqrt[3]{x^{2}}}$

Step 11

Move the limit under the radical sign.

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{{(lim_{x \to 3} x)}^{5}}} - 8 \sqrt[3]{lim_{x \to 3} x} + \frac{1}{\sqrt[3]{lim_{x \to 3} x^{2}}}$

Step 12

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{{(lim_{x \to 3} x)}^{5}}} - 8 \sqrt[3]{lim_{x \to 3} x} + \frac{1}{\sqrt[3]{{(lim_{x \to 3} x)}^{2}}}$

Step 13

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 13.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{3^{5}}} - 8 \sqrt[3]{lim_{x \to 3} x} + \frac{1}{\sqrt[3]{{(lim_{x \to 3} x)}^{2}}}$

Step 13.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{3^{5}}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{{(lim_{x \to 3} x)}^{2}}}$

Step 13.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{7}{12} \cdot \frac{1}{\sqrt[12]{3^{5}}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14

Simplify the answer.

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Step 14.1

Simplify each term.

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Step 14.1.1

Combine.

$\frac{7 \cdot 1}{12 \sqrt[12]{3^{5}}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.2

Multiply $7$ by $1$ .

$\frac{7}{12 \sqrt[12]{3^{5}}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.3

Raise $3$ to the power of $5$ .

$\frac{7}{12 \sqrt[12]{243}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.4

Multiply $\frac{7}{12 \sqrt[12]{243}}$ by $\frac{{\sqrt[12]{243}}^{11}}{{\sqrt[12]{243}}^{11}}$ .

$\frac{7}{12 \sqrt[12]{243}} \cdot \frac{{\sqrt[12]{243}}^{11}}{{\sqrt[12]{243}}^{11}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5

Combine and simplify the denominator.

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Step 14.1.5.1

Multiply $\frac{7}{12 \sqrt[12]{243}}$ by $\frac{{\sqrt[12]{243}}^{11}}{{\sqrt[12]{243}}^{11}}$ .

$\frac{7 {\sqrt[12]{243}}^{11}}{12 \sqrt[12]{243} {\sqrt[12]{243}}^{11}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.2

Move $\sqrt[12]{243}$ .

$\frac{7 {\sqrt[12]{243}}^{11}}{12 (\sqrt[12]{243} {\sqrt[12]{243}}^{11})} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.3

Raise $\sqrt[12]{243}$ to the power of $1$ .

$\frac{7 {\sqrt[12]{243}}^{11}}{12 ({\sqrt[12]{243}}^{1} {\sqrt[12]{243}}^{11})} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.4

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

$\frac{7 {\sqrt[12]{243}}^{11}}{12 {\sqrt[12]{243}}^{1 + 11}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.5

Add $1$ and $11$ .

$\frac{7 {\sqrt[12]{243}}^{11}}{12 {\sqrt[12]{243}}^{12}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.6

Rewrite ${\sqrt[12]{243}}^{12}$ as $243$ .

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Step 14.1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[12]{243}$ as $243^{\frac{1}{12}}$ .

$\frac{7 {\sqrt[12]{243}}^{11}}{12 {(243^{\frac{1}{12}})}^{12}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.6.2

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

$\frac{7 {\sqrt[12]{243}}^{11}}{12 \cdot 243^{\frac{1}{12} \cdot 12}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.6.3

Combine $\frac{1}{12}$ and $12$ .

$\frac{7 {\sqrt[12]{243}}^{11}}{12 \cdot 243^{\frac{12}{12}}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.6.4

Cancel the common factor of $12$ .

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Step 14.1.5.6.4.1

Cancel the common factor.

$\frac{7 {\sqrt[12]{243}}^{11}}{12 \cdot 243^{\frac{12}{12}}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.6.4.2

Rewrite the expression.

$\frac{7 {\sqrt[12]{243}}^{11}}{12 \cdot 243^{1}} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.5.6.5

Evaluate the exponent.

$\frac{7 {\sqrt[12]{243}}^{11}}{12 \cdot 243} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.6

Simplify the numerator.

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Step 14.1.6.1

Rewrite ${\sqrt[12]{243}}^{11}$ as $\sqrt[12]{243^{11}}$ .

$\frac{7 \sqrt[12]{243^{11}}}{12 \cdot 243} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.6.2

Raise $243$ to the power of $11$ .

$\frac{7 \sqrt[12]{174449211009120000000000000}}{12 \cdot 243} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.6.3

Rewrite $174449211009120000000000000$ as $10^{12} \cdot 174449211009120$ .

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Step 14.1.6.3.1

Factor $1000000000000$ out of $174449211009120000000000000$ .

$\frac{7 \sqrt[12]{1000000000000 (174449211009120)}}{12 \cdot 243} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.6.3.2

Rewrite $1000000000000$ as $10^{12}$ .

$\frac{7 \sqrt[12]{10^{12} \cdot 174449211009120}}{12 \cdot 243} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.6.4

Pull terms out from under the radical.

$\frac{7 \cdot 10 \sqrt[12]{174449211009120}}{12 \cdot 243} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.6.5

Multiply $7$ by $10$ .

$\frac{70 \sqrt[12]{174449211009120}}{12 \cdot 243} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.7

Multiply $12$ by $243$ .

$\frac{70 \sqrt[12]{174449211009120}}{2916} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.8

Cancel the common factor of $70$ and $2916$ .

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Step 14.1.8.1

Factor $2$ out of $70 \sqrt[12]{174449211009120}$ .

$\frac{2 (35 \sqrt[12]{174449211009120})}{2916} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.8.2

Cancel the common factors.

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Step 14.1.8.2.1

Factor $2$ out of $2916$ .

$\frac{2 (35 \sqrt[12]{174449211009120})}{2 (1458)} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.8.2.2

Cancel the common factor.

$\frac{2 (35 \sqrt[12]{174449211009120})}{2 \cdot 1458} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.8.2.3

Rewrite the expression.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{3^{2}}}$

Step 14.1.9

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

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{9}}$

Step 14.1.10

Multiply $\frac{1}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{1}{\sqrt[3]{9}} \cdot \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$

Step 14.1.11

Combine and simplify the denominator.

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Step 14.1.11.1

Multiply $\frac{1}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

Step 14.1.11.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

Step 14.1.11.3

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

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Step 14.1.11.4

Add $1$ and $2$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

Step 14.1.11.5

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

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Step 14.1.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{{(9^{\frac{1}{3}})}^{3}}$

Step 14.1.11.5.2

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

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

Step 14.1.11.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 14.1.11.5.4

Cancel the common factor of $3$ .

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Step 14.1.11.5.4.1

Cancel the common factor.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 14.1.11.5.4.2

Rewrite the expression.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9^{1}}$

Step 14.1.11.5.5

Evaluate the exponent.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{{\sqrt[3]{9}}^{2}}{9}$

Step 14.1.12

Simplify the numerator.

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Step 14.1.12.1

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{\sqrt[3]{9^{2}}}{9}$

Step 14.1.12.2

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

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{\sqrt[3]{81}}{9}$

Step 14.1.12.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 14.1.12.3.1

Factor $27$ out of $81$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{\sqrt[3]{27 (3)}}{9}$

Step 14.1.12.3.2

Rewrite $27$ as $3^{3}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{\sqrt[3]{3^{3} \cdot 3}}{9}$

Step 14.1.12.4

Pull terms out from under the radical.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{3 \sqrt[3]{3}}{9}$

Step 14.1.13

Cancel the common factor of $3$ and $9$ .

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Step 14.1.13.1

Factor $3$ out of $3 \sqrt[3]{3}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{3 (\sqrt[3]{3})}{9}$

Step 14.1.13.2

Cancel the common factors.

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Step 14.1.13.2.1

Factor $3$ out of $9$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{3 \sqrt[3]{3}}{3 \cdot 3}$

Step 14.1.13.2.2

Cancel the common factor.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{3 \sqrt[3]{3}}{3 \cdot 3}$

Step 14.1.13.2.3

Rewrite the expression.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} + \frac{\sqrt[3]{3}}{3}$

Step 14.2

To write $- 8 \sqrt[3]{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} - 8 \sqrt[3]{3} \cdot \frac{3}{3} + \frac{\sqrt[3]{3}}{3}$

Step 14.3

Combine $- 8 \sqrt[3]{3}$ and $\frac{3}{3}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} + \frac{- 8 \sqrt[3]{3} \cdot 3}{3} + \frac{\sqrt[3]{3}}{3}$

Step 14.4

Combine the numerators over the common denominator.

$\frac{35 \sqrt[12]{174449211009120}}{1458} + \frac{- 8 \sqrt[3]{3} \cdot 3 + \sqrt[3]{3}}{3}$

Step 14.5

Simplify each term.

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Step 14.5.1

Simplify the numerator.

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Step 14.5.1.1

Multiply $3$ by $- 8$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} + \frac{- 24 \sqrt[3]{3} + \sqrt[3]{3}}{3}$

Step 14.5.1.2

Add $- 24 \sqrt[3]{3}$ and $\sqrt[3]{3}$ .

$\frac{35 \sqrt[12]{174449211009120}}{1458} + \frac{- 23 \sqrt[3]{3}}{3}$

Step 14.5.2

Move the negative in front of the fraction.

$\frac{35 \sqrt[12]{174449211009120}}{1458} - \frac{23 \sqrt[3]{3}}{3}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{35 \sqrt[12]{174449211009120}}{1458} - \frac{23 \sqrt[3]{3}}{3}$

Decimal Form:

$- 10.68817030 \dots$