Calculus Examples

$lim_{x \to 8} \sqrt[4]{\frac{7 + 81 x^{2}}{x^{2} + 8}}$

Step 1

Move the limit under the radical sign.

$\sqrt[4]{lim_{x \to 8} \frac{7 + 81 x^{2}}{x^{2} + 8}}$

Step 2

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

$\sqrt[4]{\frac{lim_{x \to 8} 7 + 81 x^{2}}{lim_{x \to 8} x^{2} + 8}}$

Step 3

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

$\sqrt[4]{\frac{lim_{x \to 8} 7 + lim_{x \to 8} 81 x^{2}}{lim_{x \to 8} x^{2} + 8}}$

Step 4

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\sqrt[4]{\frac{7 + lim_{x \to 8} 81 x^{2}}{lim_{x \to 8} x^{2} + 8}}$

Step 5

Move the term $81$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[4]{\frac{7 + 81 lim_{x \to 8} x^{2}}{lim_{x \to 8} x^{2} + 8}}$

Step 6

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

$\sqrt[4]{\frac{7 + 81 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} x^{2} + 8}}$

Step 7

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

$\sqrt[4]{\frac{7 + 81 {(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} x^{2} + lim_{x \to 8} 8}}$

Step 8

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

$\sqrt[4]{\frac{7 + 81 {(lim_{x \to 8} x)}^{2}}{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 8}}$

Step 9

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

$\sqrt[4]{\frac{7 + 81 {(lim_{x \to 8} x)}^{2}}{{(lim_{x \to 8} x)}^{2} + 8}}$

Step 10

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

Tap for more steps...

Step 10.1

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

$\sqrt[4]{\frac{7 + 81 \cdot 8^{2}}{{(lim_{x \to 8} x)}^{2} + 8}}$

Step 10.2

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

$\sqrt[4]{\frac{7 + 81 \cdot 8^{2}}{8^{2} + 8}}$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

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

$\sqrt[4]{\frac{7 + 81 \cdot 64}{8^{2} + 8}}$

Step 11.2

Multiply $81$ by $64$ .

$\sqrt[4]{\frac{7 + 5184}{8^{2} + 8}}$

Step 11.3

Add $7$ and $5184$ .

$\sqrt[4]{\frac{5191}{8^{2} + 8}}$

Step 11.4

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

$\sqrt[4]{\frac{5191}{64 + 8}}$

Step 11.5

Add $64$ and $8$ .

$\sqrt[4]{\frac{5191}{72}}$

Step 11.6

Rewrite $\sqrt[4]{\frac{5191}{72}}$ as $\frac{\sqrt[4]{5191}}{\sqrt[4]{72}}$ .

$\frac{\sqrt[4]{5191}}{\sqrt[4]{72}}$

Step 11.7

Multiply $\frac{\sqrt[4]{5191}}{\sqrt[4]{72}}$ by $\frac{{\sqrt[4]{72}}^{3}}{{\sqrt[4]{72}}^{3}}$ .

$\frac{\sqrt[4]{5191}}{\sqrt[4]{72}} \cdot \frac{{\sqrt[4]{72}}^{3}}{{\sqrt[4]{72}}^{3}}$

Step 11.8

Combine and simplify the denominator.

Tap for more steps...

Step 11.8.1

Multiply $\frac{\sqrt[4]{5191}}{\sqrt[4]{72}}$ by $\frac{{\sqrt[4]{72}}^{3}}{{\sqrt[4]{72}}^{3}}$ .

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{\sqrt[4]{72} {\sqrt[4]{72}}^{3}}$

Step 11.8.2

Raise $\sqrt[4]{72}$ to the power of $1$ .

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{{\sqrt[4]{72}}^{1} {\sqrt[4]{72}}^{3}}$

Step 11.8.3

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

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{{\sqrt[4]{72}}^{1 + 3}}$

Step 11.8.4

Add $1$ and $3$ .

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{{\sqrt[4]{72}}^{4}}$

Step 11.8.5

Rewrite ${\sqrt[4]{72}}^{4}$ as $72$ .

Tap for more steps...

Step 11.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{72}$ as $72^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{{(72^{\frac{1}{4}})}^{4}}$

Step 11.8.5.2

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

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{72^{\frac{1}{4} \cdot 4}}$

Step 11.8.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{72^{\frac{4}{4}}}$

Step 11.8.5.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 11.8.5.4.1

Cancel the common factor.

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{72^{\frac{4}{4}}}$

Step 11.8.5.4.2

Rewrite the expression.

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{72^{1}}$

Step 11.8.5.5

Evaluate the exponent.

$\frac{\sqrt[4]{5191} {\sqrt[4]{72}}^{3}}{72}$

Step 11.9

Simplify the numerator.

Tap for more steps...

Step 11.9.1

Rewrite ${\sqrt[4]{72}}^{3}$ as $\sqrt[4]{72^{3}}$ .

$\frac{\sqrt[4]{5191} \sqrt[4]{72^{3}}}{72}$

Step 11.9.2

Raise $72$ to the power of $3$ .

$\frac{\sqrt[4]{5191} \sqrt[4]{373248}}{72}$

Step 11.9.3

Rewrite $373248$ as $12^{4} \cdot 18$ .

Tap for more steps...

Step 11.9.3.1

Factor $20736$ out of $373248$ .

$\frac{\sqrt[4]{5191} \sqrt[4]{20736 (18)}}{72}$

Step 11.9.3.2

Rewrite $20736$ as $12^{4}$ .

$\frac{\sqrt[4]{5191} \sqrt[4]{12^{4} \cdot 18}}{72}$

Step 11.9.4

Pull terms out from under the radical.

$\frac{\sqrt[4]{5191} \cdot 12 \sqrt[4]{18}}{72}$

Step 11.9.5

Combine exponents.

Tap for more steps...

Step 11.9.5.1

Combine using the product rule for radicals.

$\frac{12 \sqrt[4]{5191 \cdot 18}}{72}$

Step 11.9.5.2

Multiply $5191$ by $18$ .

$\frac{12 \sqrt[4]{93438}}{72}$

Step 11.10

Cancel the common factor of $12$ and $72$ .

Tap for more steps...

Step 11.10.1

Factor $12$ out of $12 \sqrt[4]{93438}$ .

$\frac{12 (\sqrt[4]{93438})}{72}$

Step 11.10.2

Cancel the common factors.

Tap for more steps...

Step 11.10.2.1

Factor $12$ out of $72$ .

$\frac{12 \sqrt[4]{93438}}{12 \cdot 6}$

Step 11.10.2.2

Cancel the common factor.

$\frac{12 \sqrt[4]{93438}}{12 \cdot 6}$

Step 11.10.2.3

Rewrite the expression.

$\frac{\sqrt[4]{93438}}{6}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{93438}}{6}$

Decimal Form:

$2.91393347 \dots$