Calculus Examples

$lim_{x \to 8} \sqrt[3]{\frac{6 + 9 x + 3 x^{2}}{1 + 8 x^{2}}}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to 8} \frac{6 + 9 x + 3 x^{2}}{1 + 8 x^{2}}}$

Step 2

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

$\sqrt[3]{\frac{lim_{x \to 8} 6 + 9 x + 3 x^{2}}{lim_{x \to 8} 1 + 8 x^{2}}}$

Step 3

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

$\sqrt[3]{\frac{lim_{x \to 8} 6 + lim_{x \to 8} 9 x + lim_{x \to 8} 3 x^{2}}{lim_{x \to 8} 1 + 8 x^{2}}}$

Step 4

Evaluate the limit of $6$ which is constant as $x$ approaches $8$ .

$\sqrt[3]{\frac{6 + lim_{x \to 8} 9 x + lim_{x \to 8} 3 x^{2}}{lim_{x \to 8} 1 + 8 x^{2}}}$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{6 + 9 lim_{x \to 8} x + lim_{x \to 8} 3 x^{2}}{lim_{x \to 8} 1 + 8 x^{2}}}$

Step 6

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

$\sqrt[3]{\frac{6 + 9 lim_{x \to 8} x + 3 lim_{x \to 8} x^{2}}{lim_{x \to 8} 1 + 8 x^{2}}}$

Step 7

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

$\sqrt[3]{\frac{6 + 9 lim_{x \to 8} x + 3 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 1 + 8 x^{2}}}$

Step 8

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

$\sqrt[3]{\frac{6 + 9 lim_{x \to 8} x + 3 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 1 + lim_{x \to 8} 8 x^{2}}}$

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\sqrt[3]{\frac{6 + 9 lim_{x \to 8} x + 3 {(lim_{x \to 8} x)}^{2}}{1 + lim_{x \to 8} 8 x^{2}}}$

Step 10

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

$\sqrt[3]{\frac{6 + 9 lim_{x \to 8} x + 3 {(lim_{x \to 8} x)}^{2}}{1 + 8 lim_{x \to 8} x^{2}}}$

Step 11

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

$\sqrt[3]{\frac{6 + 9 lim_{x \to 8} x + 3 {(lim_{x \to 8} x)}^{2}}{1 + 8 {(lim_{x \to 8} x)}^{2}}}$

Step 12

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

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Step 12.1

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

$\sqrt[3]{\frac{6 + 9 \cdot 8 + 3 {(lim_{x \to 8} x)}^{2}}{1 + 8 {(lim_{x \to 8} x)}^{2}}}$

Step 12.2

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

$\sqrt[3]{\frac{6 + 9 \cdot 8 + 3 \cdot 8^{2}}{1 + 8 {(lim_{x \to 8} x)}^{2}}}$

Step 12.3

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

$\sqrt[3]{\frac{6 + 9 \cdot 8 + 3 \cdot 8^{2}}{1 + 8 \cdot 8^{2}}}$

Step 13

Simplify the answer.

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Step 13.1

Multiply $9$ by $8$ .

$\sqrt[3]{\frac{6 + 72 + 3 \cdot 8^{2}}{1 + 8 \cdot 8^{2}}}$

Step 13.2

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

$\sqrt[3]{\frac{6 + 72 + 3 \cdot 64}{1 + 8 \cdot 8^{2}}}$

Step 13.3

Multiply $3$ by $64$ .

$\sqrt[3]{\frac{6 + 72 + 192}{1 + 8 \cdot 8^{2}}}$

Step 13.4

Add $6$ and $72$ .

$\sqrt[3]{\frac{78 + 192}{1 + 8 \cdot 8^{2}}}$

Step 13.5

Add $78$ and $192$ .

$\sqrt[3]{\frac{270}{1 + 8 \cdot 8^{2}}}$

Step 13.6

Multiply $8$ by $8^{2}$ by adding the exponents.

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Step 13.6.1

Multiply $8$ by $8^{2}$ .

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Step 13.6.1.1

Raise $8$ to the power of $1$ .

$\sqrt[3]{\frac{270}{1 + 8^{1} \cdot 8^{2}}}$

Step 13.6.1.2

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

$\sqrt[3]{\frac{270}{1 + 8^{1 + 2}}}$

Step 13.6.2

Add $1$ and $2$ .

$\sqrt[3]{\frac{270}{1 + 8^{3}}}$

Step 13.7

Raise $8$ to the power of $3$ .

$\sqrt[3]{\frac{270}{1 + 512}}$

Step 13.8

Add $1$ and $512$ .

$\sqrt[3]{\frac{270}{513}}$

Step 13.9

Cancel the common factor of $270$ and $513$ .

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Step 13.9.1

Factor $27$ out of $270$ .

$\sqrt[3]{\frac{27 (10)}{513}}$

Step 13.9.2

Cancel the common factors.

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Step 13.9.2.1

Factor $27$ out of $513$ .

$\sqrt[3]{\frac{27 \cdot 10}{27 \cdot 19}}$

Step 13.9.2.2

Cancel the common factor.

$\sqrt[3]{\frac{27 \cdot 10}{27 \cdot 19}}$

Step 13.9.2.3

Rewrite the expression.

$\sqrt[3]{\frac{10}{19}}$

Step 13.10

Rewrite $\sqrt[3]{\frac{10}{19}}$ as $\frac{\sqrt[3]{10}}{\sqrt[3]{19}}$ .

$\frac{\sqrt[3]{10}}{\sqrt[3]{19}}$

Step 13.11

Multiply $\frac{\sqrt[3]{10}}{\sqrt[3]{19}}$ by $\frac{{\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{2}}$ .

$\frac{\sqrt[3]{10}}{\sqrt[3]{19}} \cdot \frac{{\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{2}}$

Step 13.12

Combine and simplify the denominator.

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Step 13.12.1

Multiply $\frac{\sqrt[3]{10}}{\sqrt[3]{19}}$ by $\frac{{\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{2}}$ .

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{\sqrt[3]{19} {\sqrt[3]{19}}^{2}}$

Step 13.12.2

Raise $\sqrt[3]{19}$ to the power of $1$ .

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{1} {\sqrt[3]{19}}^{2}}$

Step 13.12.3

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

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{1 + 2}}$

Step 13.12.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{{\sqrt[3]{19}}^{3}}$

Step 13.12.5

Rewrite ${\sqrt[3]{19}}^{3}$ as $19$ .

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Step 13.12.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{19}$ as $19^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{{(19^{\frac{1}{3}})}^{3}}$

Step 13.12.5.2

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

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{19^{\frac{1}{3} \cdot 3}}$

Step 13.12.5.3

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

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{19^{\frac{3}{3}}}$

Step 13.12.5.4

Cancel the common factor of $3$ .

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Step 13.12.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{19^{\frac{3}{3}}}$

Step 13.12.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{19^{1}}$

Step 13.12.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{10} {\sqrt[3]{19}}^{2}}{19}$

Step 13.13

Simplify the numerator.

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Step 13.13.1

Rewrite ${\sqrt[3]{19}}^{2}$ as $\sqrt[3]{19^{2}}$ .

$\frac{\sqrt[3]{10} \sqrt[3]{19^{2}}}{19}$

Step 13.13.2

Raise $19$ to the power of $2$ .

$\frac{\sqrt[3]{10} \sqrt[3]{361}}{19}$

Step 13.14

Simplify the numerator.

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Step 13.14.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{10 \cdot 361}}{19}$

Step 13.14.2

Multiply $10$ by $361$ .

$\frac{\sqrt[3]{3610}}{19}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{3610}}{19}$

Decimal Form:

$0.80738770 \dots$