Calculus Examples

$lim_{x \to 8} \sqrt[3]{\frac{3 + 6 x + 5 x^{2}}{1 + 64 x^{2}}}$

Step 1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to 8} \frac{3 + 6 x + 5 x^{2}}{1 + 64 x^{2}}}$

Step 2

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

$\sqrt[3]{\frac{lim_{x \to 8} 3 + 6 x + 5 x^{2}}{lim_{x \to 8} 1 + 64 x^{2}}}$

Step 3

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

$\sqrt[3]{\frac{lim_{x \to 8} 3 + lim_{x \to 8} 6 x + lim_{x \to 8} 5 x^{2}}{lim_{x \to 8} 1 + 64 x^{2}}}$

Step 4

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\sqrt[3]{\frac{3 + lim_{x \to 8} 6 x + lim_{x \to 8} 5 x^{2}}{lim_{x \to 8} 1 + 64 x^{2}}}$

Step 5

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

$\sqrt[3]{\frac{3 + 6 lim_{x \to 8} x + lim_{x \to 8} 5 x^{2}}{lim_{x \to 8} 1 + 64 x^{2}}}$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{3 + 6 lim_{x \to 8} x + 5 lim_{x \to 8} x^{2}}{lim_{x \to 8} 1 + 64 x^{2}}}$

Step 7

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

$\sqrt[3]{\frac{3 + 6 lim_{x \to 8} x + 5 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 1 + 64 x^{2}}}$

Step 8

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

$\sqrt[3]{\frac{3 + 6 lim_{x \to 8} x + 5 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} 1 + lim_{x \to 8} 64 x^{2}}}$

Step 9

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

$\sqrt[3]{\frac{3 + 6 lim_{x \to 8} x + 5 {(lim_{x \to 8} x)}^{2}}{1 + lim_{x \to 8} 64 x^{2}}}$

Step 10

Move the term $64$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{\frac{3 + 6 lim_{x \to 8} x + 5 {(lim_{x \to 8} x)}^{2}}{1 + 64 lim_{x \to 8} x^{2}}}$

Step 11

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

$\sqrt[3]{\frac{3 + 6 lim_{x \to 8} x + 5 {(lim_{x \to 8} x)}^{2}}{1 + 64 {(lim_{x \to 8} x)}^{2}}}$

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 $x$ .

$\sqrt[3]{\frac{3 + 6 \cdot 8 + 5 {(lim_{x \to 8} x)}^{2}}{1 + 64 {(lim_{x \to 8} x)}^{2}}}$

Step 12.2

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

$\sqrt[3]{\frac{3 + 6 \cdot 8 + 5 \cdot 8^{2}}{1 + 64 {(lim_{x \to 8} x)}^{2}}}$

Step 12.3

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

$\sqrt[3]{\frac{3 + 6 \cdot 8 + 5 \cdot 8^{2}}{1 + 64 \cdot 8^{2}}}$

Step 13

Simplify the answer.

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Step 13.1

Multiply $6$ by $8$ .

$\sqrt[3]{\frac{3 + 48 + 5 \cdot 8^{2}}{1 + 64 \cdot 8^{2}}}$

Step 13.2

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

$\sqrt[3]{\frac{3 + 48 + 5 \cdot 64}{1 + 64 \cdot 8^{2}}}$

Step 13.3

Multiply $5$ by $64$ .

$\sqrt[3]{\frac{3 + 48 + 320}{1 + 64 \cdot 8^{2}}}$

Step 13.4

Add $3$ and $48$ .

$\sqrt[3]{\frac{51 + 320}{1 + 64 \cdot 8^{2}}}$

Step 13.5

Add $51$ and $320$ .

$\sqrt[3]{\frac{371}{1 + 64 \cdot 8^{2}}}$

Step 13.6

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

$\sqrt[3]{\frac{371}{1 + 64 \cdot 64}}$

Step 13.7

Multiply $64$ by $64$ .

$\sqrt[3]{\frac{371}{1 + 4096}}$

Step 13.8

Add $1$ and $4096$ .

$\sqrt[3]{\frac{371}{4097}}$

Step 13.9

Rewrite $\sqrt[3]{\frac{371}{4097}}$ as $\frac{\sqrt[3]{371}}{\sqrt[3]{4097}}$ .

$\frac{\sqrt[3]{371}}{\sqrt[3]{4097}}$

Step 13.10

Multiply $\frac{\sqrt[3]{371}}{\sqrt[3]{4097}}$ by $\frac{{\sqrt[3]{4097}}^{2}}{{\sqrt[3]{4097}}^{2}}$ .

$\frac{\sqrt[3]{371}}{\sqrt[3]{4097}} \cdot \frac{{\sqrt[3]{4097}}^{2}}{{\sqrt[3]{4097}}^{2}}$

Step 13.11

Combine and simplify the denominator.

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Step 13.11.1

Multiply $\frac{\sqrt[3]{371}}{\sqrt[3]{4097}}$ by $\frac{{\sqrt[3]{4097}}^{2}}{{\sqrt[3]{4097}}^{2}}$ .

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{\sqrt[3]{4097} {\sqrt[3]{4097}}^{2}}$

Step 13.11.2

Raise $\sqrt[3]{4097}$ to the power of $1$ .

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{{\sqrt[3]{4097}}^{1} {\sqrt[3]{4097}}^{2}}$

Step 13.11.3

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

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{{\sqrt[3]{4097}}^{1 + 2}}$

Step 13.11.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{{\sqrt[3]{4097}}^{3}}$

Step 13.11.5

Rewrite ${\sqrt[3]{4097}}^{3}$ as $4097$ .

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Step 13.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4097}$ as $4097^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{{(4097^{\frac{1}{3}})}^{3}}$

Step 13.11.5.2

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

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{4097^{\frac{1}{3} \cdot 3}}$

Step 13.11.5.3

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

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{4097^{\frac{3}{3}}}$

Step 13.11.5.4

Cancel the common factor of $3$ .

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Step 13.11.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{4097^{\frac{3}{3}}}$

Step 13.11.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{4097^{1}}$

Step 13.11.5.5

Evaluate the exponent.

$\frac{\sqrt[3]{371} {\sqrt[3]{4097}}^{2}}{4097}$

Step 13.12

Simplify the numerator.

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Step 13.12.1

Rewrite ${\sqrt[3]{4097}}^{2}$ as $\sqrt[3]{4097^{2}}$ .

$\frac{\sqrt[3]{371} \sqrt[3]{4097^{2}}}{4097}$

Step 13.12.2

Raise $4097$ to the power of $2$ .

$\frac{\sqrt[3]{371} \sqrt[3]{16785409}}{4097}$

Step 13.13

Simplify the numerator.

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Step 13.13.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{371 \cdot 16785409}}{4097}$

Step 13.13.2

Multiply $371$ by $16785409$ .

$\frac{\sqrt[3]{6227386739}}{4097}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{6227386739}}{4097}$

Decimal Form:

$0.44905821 \dots$