Approximate the following number using a calculator.

(Round to three decimal places as needed.)

Ask by Franklin Horton. in the United States

Mar 26,2025

Question

Approximate the following number using a calculator. $$ 7.9\left(\frac{1}{5}\right)^{2.2} $$ $$ 7.9\left(\frac{1}{5}\right)^{2.2} \approx \square $$ (Round to three decimal places as needed.)

UpStudy AI · Accepted Answer

We start with the expression

$$
7.9\left(\frac{1}{5}\right)^{2.2}.
$$

First, rewrite the expression using the property $$\left(\frac{1}{5}\right)^{2.2} = 5^{-2.2}$$:

$$
7.9\left(\frac{1}{5}\right)^{2.2} = 7.9 \cdot 5^{-2.2}.
$$

Next, we recognize that

$$
5^{-2.2} = \exp\left(-2.2 \ln 5\right).
$$

Calculate $$-2.2 \ln 5$$. The natural logarithm of 5 is approximately

$$
\ln 5 \approx 1.60944,
$$

so

$$
-2.2 \ln 5 \approx -2.2 \times 1.60944 \approx -3.54077.
$$

Now, compute the exponential:

$$
5^{-2.2} = \exp(-3.54077) \approx 0.02907.
$$

Finally, multiply this by 7.9:

$$
7.9\left(\frac{1}{5}\right)^{2.2} \approx 7.9 \times 0.02907 \approx 0.229.
$$

Thus, the rounded result is

$$
7.9\left(\frac{1}{5}\right)^{2.2} \approx 0.229.
$$
$$ 7.9\left(\frac{1}{5}\right)^{2.2} \approx 0.229 $$

Approximate the following number using a calculator.

(Round to three decimal places as needed.)

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Approximate the following number using a calculator. (Round to three decimal places as needed.)