Calculus Examples

$lim_{x \to 8} \sqrt[5]{\frac{19 x^{7} + x^{10} - 73}{32 x^{10} - 17^{8} + 42}}$

Step 1

Move the limit under the radical sign.

$\sqrt[5]{lim_{x \to 8} \frac{19 x^{7} + x^{10} - 73}{32 x^{10} - 17^{8} + 42}}$

Step 2

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

$\sqrt[5]{\frac{lim_{x \to 8} 19 x^{7} + x^{10} - 73}{lim_{x \to 8} 32 x^{10} - 17^{8} + 42}}$

Step 3

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

$\sqrt[5]{\frac{lim_{x \to 8} 19 x^{7} + lim_{x \to 8} x^{10} - lim_{x \to 8} 73}{lim_{x \to 8} 32 x^{10} - 17^{8} + 42}}$

Step 4

Move the term $19$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[5]{\frac{19 lim_{x \to 8} x^{7} + lim_{x \to 8} x^{10} - lim_{x \to 8} 73}{lim_{x \to 8} 32 x^{10} - 17^{8} + 42}}$

Step 5

Move the exponent $7$ from $x^{7}$ outside the limit using the Limits Power Rule.

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + lim_{x \to 8} x^{10} - lim_{x \to 8} 73}{lim_{x \to 8} 32 x^{10} - 17^{8} + 42}}$

Step 6

Move the exponent $10$ from $x^{10}$ outside the limit using the Limits Power Rule.

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + {(lim_{x \to 8} x)}^{10} - lim_{x \to 8} 73}{lim_{x \to 8} 32 x^{10} - 17^{8} + 42}}$

Step 7

Evaluate the limit of $73$ which is constant as $x$ approaches $8$ .

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + {(lim_{x \to 8} x)}^{10} - 1 \cdot 73}{lim_{x \to 8} 32 x^{10} - 17^{8} + 42}}$

Step 8

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

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + {(lim_{x \to 8} x)}^{10} - 1 \cdot 73}{lim_{x \to 8} 32 x^{10} - lim_{x \to 8} 17^{8} + lim_{x \to 8} 42}}$

Step 9

Move the term $32$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + {(lim_{x \to 8} x)}^{10} - 1 \cdot 73}{32 lim_{x \to 8} x^{10} - lim_{x \to 8} 17^{8} + lim_{x \to 8} 42}}$

Step 10

Move the exponent $10$ from $x^{10}$ outside the limit using the Limits Power Rule.

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + {(lim_{x \to 8} x)}^{10} - 1 \cdot 73}{32 {(lim_{x \to 8} x)}^{10} - lim_{x \to 8} 17^{8} + lim_{x \to 8} 42}}$

Step 11

Evaluate the limit of $17^{8}$ which is constant as $x$ approaches $8$ .

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + {(lim_{x \to 8} x)}^{10} - 1 \cdot 73}{32 {(lim_{x \to 8} x)}^{10} - 17^{8} + lim_{x \to 8} 42}}$

Step 12

Evaluate the limit of $42$ which is constant as $x$ approaches $8$ .

$\sqrt[5]{\frac{19 {(lim_{x \to 8} x)}^{7} + {(lim_{x \to 8} x)}^{10} - 1 \cdot 73}{32 {(lim_{x \to 8} x)}^{10} - 17^{8} + 42}}$

Step 13

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

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Step 13.1

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

$\sqrt[5]{\frac{19 \cdot 8^{7} + {(lim_{x \to 8} x)}^{10} - 1 \cdot 73}{32 {(lim_{x \to 8} x)}^{10} - 17^{8} + 42}}$

Step 13.2

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

$\sqrt[5]{\frac{19 \cdot 8^{7} + 8^{10} - 1 \cdot 73}{32 {(lim_{x \to 8} x)}^{10} - 17^{8} + 42}}$

Step 13.3

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

$\sqrt[5]{\frac{19 \cdot 8^{7} + 8^{10} - 1 \cdot 73}{32 \cdot 8^{10} - 17^{8} + 42}}$

Step 14

Simplify the answer.

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Step 14.1

Raise $8$ to the power of $7$ .

$\sqrt[5]{\frac{19 \cdot 2097152 + 8^{10} - 1 \cdot 73}{32 \cdot 8^{10} - 17^{8} + 42}}$

Step 14.2

Multiply $19$ by $2097152$ .

$\sqrt[5]{\frac{39845888 + 8^{10} - 1 \cdot 73}{32 \cdot 8^{10} - 17^{8} + 42}}$

Step 14.3

Raise $8$ to the power of $10$ .

$\sqrt[5]{\frac{39845888 + 1073741824 - 1 \cdot 73}{32 \cdot 8^{10} - 17^{8} + 42}}$

Step 14.4

Multiply $- 1$ by $73$ .

$\sqrt[5]{\frac{39845888 + 1073741824 - 73}{32 \cdot 8^{10} - 17^{8} + 42}}$

Step 14.5

Add $39845888$ and $1073741824$ .

$\sqrt[5]{\frac{1113587712 - 73}{32 \cdot 8^{10} - 17^{8} + 42}}$

Step 14.6

Subtract $73$ from $1113587712$ .

$\sqrt[5]{\frac{1113587639}{32 \cdot 8^{10} - 17^{8} + 42}}$

Step 14.7

Raise $8$ to the power of $10$ .

$\sqrt[5]{\frac{1113587639}{32 \cdot 1073741824 - 17^{8} + 42}}$

Step 14.8

Multiply $32$ by $1073741824$ .

$\sqrt[5]{\frac{1113587639}{34359738368 - 17^{8} + 42}}$

Step 14.9

Raise $17$ to the power of $8$ .

$\sqrt[5]{\frac{1113587639}{34359738368 - 1 \cdot 6975757441 + 42}}$

Step 14.10

Multiply $- 1$ by $6975757441$ .

$\sqrt[5]{\frac{1113587639}{34359738368 - 6975757441 + 42}}$

Step 14.11

Subtract $6975757441$ from $34359738368$ .

$\sqrt[5]{\frac{1113587639}{27383980927 + 42}}$

Step 14.12

Add $27383980927$ and $42$ .

$\sqrt[5]{\frac{1113587639}{27383980969}}$

Step 14.13

Rewrite $\sqrt[5]{\frac{1113587639}{27383980969}}$ as $\frac{\sqrt[5]{1113587639}}{\sqrt[5]{27383980969}}$ .

$\frac{\sqrt[5]{1113587639}}{\sqrt[5]{27383980969}}$

Step 14.14

Multiply $\frac{\sqrt[5]{1113587639}}{\sqrt[5]{27383980969}}$ by $\frac{{\sqrt[5]{27383980969}}^{4}}{{\sqrt[5]{27383980969}}^{4}}$ .

$\frac{\sqrt[5]{1113587639}}{\sqrt[5]{27383980969}} \cdot \frac{{\sqrt[5]{27383980969}}^{4}}{{\sqrt[5]{27383980969}}^{4}}$

Step 14.15

Combine and simplify the denominator.

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Step 14.15.1

Multiply $\frac{\sqrt[5]{1113587639}}{\sqrt[5]{27383980969}}$ by $\frac{{\sqrt[5]{27383980969}}^{4}}{{\sqrt[5]{27383980969}}^{4}}$ .

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{\sqrt[5]{27383980969} {\sqrt[5]{27383980969}}^{4}}$

Step 14.15.2

Raise $\sqrt[5]{27383980969}$ to the power of $1$ .

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{{\sqrt[5]{27383980969}}^{1} {\sqrt[5]{27383980969}}^{4}}$

Step 14.15.3

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

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{{\sqrt[5]{27383980969}}^{1 + 4}}$

Step 14.15.4

Add $1$ and $4$ .

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{{\sqrt[5]{27383980969}}^{5}}$

Step 14.15.5

Rewrite ${\sqrt[5]{27383980969}}^{5}$ as $27383980969$ .

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Step 14.15.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{27383980969}$ as $27383980969^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{{(27383980969^{\frac{1}{5}})}^{5}}$

Step 14.15.5.2

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

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{27383980969^{\frac{1}{5} \cdot 5}}$

Step 14.15.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{27383980969^{\frac{5}{5}}}$

Step 14.15.5.4

Cancel the common factor of $5$ .

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Step 14.15.5.4.1

Cancel the common factor.

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{27383980969^{\frac{5}{5}}}$

Step 14.15.5.4.2

Rewrite the expression.

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{27383980969^{1}}$

Step 14.15.5.5

Evaluate the exponent.

$\frac{\sqrt[5]{1113587639} {\sqrt[5]{27383980969}}^{4}}{27383980969}$

Step 14.16

Rewrite ${\sqrt[5]{27383980969}}^{4}$ as $\sqrt[5]{27383980969^{4}}$ .

$\frac{\sqrt[5]{1113587639} \sqrt[5]{27383980969^{4}}}{27383980969}$

Step 14.17

Combine using the product rule for radicals.

$\frac{\sqrt[5]{1113587639 \cdot 27383980969^{4}}}{27383980969}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[5]{1113587639 \cdot 27383980969^{4}}}{27383980969}$

Decimal Form:

$0.52704241 \dots$