$$ 1 \leftarrow \begin{array}{llll} ext { Approximate each number using a calculator. } \ \begin{array}{lll} ext { (a) } 2.7^{3.1} & ext { (b) } 2.71^{3.14} & ext { (c) } 2.718^{3.141}\end{array} & ext { (d) } e^{\pi}\end{array} $$

Question

$$ 1 \leftarrow \begin{array}{llll}	ext { Approximate each number using a calculator. } \ \begin{array}{lll}	ext { (a) } 2.7^{3.1} & 	ext { (b) } 2.71^{3.14} & 	ext { (c) } 2.718^{3.141}\end{array} & 	ext { (d) } e^{\pi}\end{array} $$

UpStudy AI · Accepted Answer

Let's approximate each of the given expressions step by step:

1. **(a)** $$ 2.7^{3.1} $$
2. **(b)** $$ 2.71^{3.14} $$
3. **(c)** $$ 2.718^{3.141} $$
4. **(d)** $$ e^{\pi} $$

I will calculate these values using a calculator. Let's proceed with the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$e^{\pi }$$
Calculate or simplify the expression $$ 2.71^{3.14} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2.71^{3.14}$$
- step1: Convert the expressions:
  $$\left(\frac{271}{100}ight)^{\frac{157}{50}}$$
- step2: Use the properties of exponents:
  $$\frac{271^{\frac{157}{50}}}{100^{\frac{157}{50}}}$$
- step3: Rewrite the expression:
  $$\frac{271^{3}\sqrt[50]{271^{7}}}{10^{6}\sqrt[25]{10^{7}}}$$
- step4: Multiply by the Conjugate:
  $$\frac{271^{3}\sqrt[50]{271^{7}}	imes \sqrt[25]{10^{18}}}{10^{6}\sqrt[25]{10^{7}}	imes \sqrt[25]{10^{18}}}$$
- step5: Multiply the numbers:
  $$\frac{271^{3}\sqrt[50]{271^{7}	imes 10^{36}}}{10^{6}\sqrt[25]{10^{7}}	imes \sqrt[25]{10^{18}}}$$
- step6: Multiply the numbers:
  $$\frac{271^{3}\sqrt[50]{271^{7}	imes 10^{36}}}{10^{7}}$$
Calculate or simplify the expression $$ 2.718^{3.141} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2.718^{3.141}$$
- step1: Convert the expressions:
  $$\left(\frac{1359}{500}ight)^{\frac{3141}{1000}}$$
- step2: Use the properties of exponents:
  $$\frac{1359^{\frac{3141}{1000}}}{500^{\frac{3141}{1000}}}$$
- step3: Rewrite the expression:
  $$\frac{1359^{3}\sqrt[1000]{1359^{141}}}{500^{3}\sqrt[1000]{500^{141}}}$$
- step4: Multiply by the Conjugate:
  $$\frac{1359^{3}\sqrt[1000]{1359^{141}}	imes \sqrt[1000]{500^{859}}}{500^{3}\sqrt[1000]{500^{141}}	imes \sqrt[1000]{500^{859}}}$$
- step5: Multiply the numbers:
  $$\frac{1359^{3}\sqrt[1000]{1359^{141}	imes 500^{859}}}{500^{3}\sqrt[1000]{500^{141}}	imes \sqrt[1000]{500^{859}}}$$
- step6: Multiply the numbers:
  $$\frac{1359^{3}\sqrt[1000]{1359^{141}	imes 500^{859}}}{500^{4}}$$
Calculate or simplify the expression $$ 2.7^{3.1} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2.7^{3.1}$$
- step1: Convert the expressions:
  $$\left(\frac{27}{10}ight)^{\frac{31}{10}}$$
- step2: Use the properties of exponents:
  $$\frac{27^{\frac{31}{10}}}{10^{\frac{31}{10}}}$$
- step3: Rewrite the expression:
  $$\frac{27^{\frac{31}{10}}}{1000\sqrt[10]{10}}$$
- step4: Rewrite the expression:
  $$\frac{19683\sqrt[10]{27}}{1000\sqrt[10]{10}}$$
- step5: Multiply by the Conjugate:
  $$\frac{19683\sqrt[10]{27}	imes \sqrt[10]{10^{9}}}{1000\sqrt[10]{10}	imes \sqrt[10]{10^{9}}}$$
- step6: Multiply the numbers:
  $$\frac{19683\sqrt[10]{27	imes 10^{9}}}{1000\sqrt[10]{10}	imes \sqrt[10]{10^{9}}}$$
- step7: Multiply the numbers:
  $$\frac{19683\sqrt[10]{27	imes 10^{9}}}{10000}$$
Here are the approximations for each expression:

1. **(a)** $$ 2.7^{3.1} \approx 21.738405 $$
2. **(b)** $$ 2.71^{3.14} \approx 22.883559 $$
3. **(c)** $$ 2.718^{3.141} \approx 23.119452 $$
4. **(d)** $$ e^{\pi} \approx 23.140693 $$

These values provide the approximations for each of the given expressions.
The approximations are: - (a) $$ 2.7^{3.1} \approx 21.738 $$ - (b) $$ 2.71^{3.14} \approx 22.884 $$ - (c) $$ 2.718^{3.141} \approx 23.119 $$ - (d) $$ e^{\pi} \approx 23.141 $$

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