Calculus Examples

$lim_{x \to 8} \sqrt[3]{\frac{3 x^{2} + 5}{6 x - 8}}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to 8} \frac{3 x^{2} + 5}{6 x - 8}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\sqrt[3]{\frac{lim_{x \to 8} 3 x^{2} + 5}{lim_{x \to 8} 6 x - 8}}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\sqrt[3]{\frac{lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 5}{lim_{x \to 8} 6 x - 8}}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{3 lim_{x \to 8} x^{2} + lim_{x \to 8} 5}{lim_{x \to 8} 6 x - 8}}$

Step 5

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

$\sqrt[3]{\frac{3 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 5}{lim_{x \to 8} 6 x - 8}}$

Step 6

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\sqrt[3]{\frac{3 {(lim_{x \to 8} x)}^{2} + 5}{lim_{x \to 8} 6 x - 8}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\sqrt[3]{\frac{3 {(lim_{x \to 8} x)}^{2} + 5}{lim_{x \to 8} 6 x - lim_{x \to 8} 8}}$

Step 8

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{3 {(lim_{x \to 8} x)}^{2} + 5}{6 lim_{x \to 8} x - lim_{x \to 8} 8}}$

Step 9

Evaluate the limit of $8$ which is constant as $x$ approaches $8$ .

$\sqrt[3]{\frac{3 {(lim_{x \to 8} x)}^{2} + 5}{6 lim_{x \to 8} x - 1 \cdot 8}}$

Step 10

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sqrt[3]{\frac{3 \cdot 8^{2} + 5}{6 lim_{x \to 8} x - 1 \cdot 8}}$

Step 10.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\sqrt[3]{\frac{3 \cdot 8^{2} + 5}{6 \cdot 8 - 1 \cdot 8}}$

Step 11

Simplify the answer.

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Step 11.1

Raise $8$ to the power of $2$ .

$\sqrt[3]{\frac{3 \cdot 64 + 5}{6 \cdot 8 - 1 \cdot 8}}$

Step 11.2

Multiply $3$ by $64$ .

$\sqrt[3]{\frac{192 + 5}{6 \cdot 8 - 1 \cdot 8}}$

Step 11.3

Add $192$ and $5$ .

$\sqrt[3]{\frac{197}{6 \cdot 8 - 1 \cdot 8}}$

Step 11.4

Multiply $6$ by $8$ .

$\sqrt[3]{\frac{197}{48 - 1 \cdot 8}}$

Step 11.5

Multiply $- 1$ by $8$ .

$\sqrt[3]{\frac{197}{48 - 8}}$

Step 11.6

Subtract $8$ from $48$ .

$\sqrt[3]{\frac{197}{40}}$

Step 11.7

Rewrite $\sqrt[3]{\frac{197}{40}}$ as $\frac{\sqrt[3]{197}}{\sqrt[3]{40}}$ .

$\frac{\sqrt[3]{197}}{\sqrt[3]{40}}$

Step 11.8

Simplify the denominator.

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Step 11.8.1

Rewrite $40$ as $2^{3} \cdot 5$ .

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Step 11.8.1.1

Factor $8$ out of $40$ .

$\frac{\sqrt[3]{197}}{\sqrt[3]{8 (5)}}$

Step 11.8.1.2

Rewrite $8$ as $2^{3}$ .

$\frac{\sqrt[3]{197}}{\sqrt[3]{2^{3} \cdot 5}}$

Step 11.8.2

Pull terms out from under the radical.

$\frac{\sqrt[3]{197}}{2 \sqrt[3]{5}}$

Step 11.9

Multiply $\frac{\sqrt[3]{197}}{2 \sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{\sqrt[3]{197}}{2 \sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$

Step 11.10

Combine and simplify the denominator.

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Step 11.10.1

Multiply $\frac{\sqrt[3]{197}}{2 \sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 \sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Step 11.10.2

Move $\sqrt[3]{5}$ .

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 (\sqrt[3]{5} {\sqrt[3]{5}}^{2})}$

Step 11.10.3

Raise $\sqrt[3]{5}$ to the power of $1$ .

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 ({\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2})}$

Step 11.10.4

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

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 {\sqrt[3]{5}}^{1 + 2}}$

Step 11.10.5

Add $1$ and $2$ .

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 {\sqrt[3]{5}}^{3}}$

Step 11.10.6

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

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Step 11.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 {(5^{\frac{1}{3}})}^{3}}$

Step 11.10.6.2

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

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{\frac{1}{3} \cdot 3}}$

Step 11.10.6.3

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

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{\frac{3}{3}}}$

Step 11.10.6.4

Cancel the common factor of $3$ .

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Step 11.10.6.4.1

Cancel the common factor.

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{\frac{3}{3}}}$

Step 11.10.6.4.2

Rewrite the expression.

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 \cdot 5^{1}}$

Step 11.10.6.5

Evaluate the exponent.

$\frac{\sqrt[3]{197} {\sqrt[3]{5}}^{2}}{2 \cdot 5}$

Step 11.11

Simplify the numerator.

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Step 11.11.1

Rewrite ${\sqrt[3]{5}}^{2}$ as $\sqrt[3]{5^{2}}$ .

$\frac{\sqrt[3]{197} \sqrt[3]{5^{2}}}{2 \cdot 5}$

Step 11.11.2

Raise $5$ to the power of $2$ .

$\frac{\sqrt[3]{197} \sqrt[3]{25}}{2 \cdot 5}$

Step 11.12

Multiply $2$ by $5$ .

$\frac{\sqrt[3]{197} \sqrt[3]{25}}{10}$

Step 11.13

Simplify the numerator.

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Step 11.13.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{197 \cdot 25}}{10}$

Step 11.13.2

Multiply $197$ by $25$ .

$\frac{\sqrt[3]{4925}}{10}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{4925}}{10}$

Decimal Form:

$1.70138295 \dots$