91Ó°ÊÓ

Powers and exponents are fundamental concepts in mathematics. When we talk about exponents, we're referring to the number of times a number, known as the base, is multiplied by itself. This is written as \(a^b\), where \(a\) is the base and \(b\) is the exponent. For example, in the expression \(1.8^4\), 1.8 is the base and 4 is the exponent. This means we multiply 1.8 by itself four times:
\[1.8 \times 1.8 \times 1.8 \times 1.8 = 10.4976\]
This allows expressions and calculations to be more concise and easier to manage, especially when working with large numbers.Exponents can also be fractions or negative numbers. A fractional exponent like \(a^{0.5}\) represents a root, such as the square root of \(a\). A negative exponent, like \(a^{-b}\), indicates that the base is taken as the reciprocal: \(\frac{1}{a^b}\). Understanding how to manipulate these is crucial for solving many problems in math.

Function evaluation involves solving a function for given input values. In our exercise, the function is \(r(w) = 1.8^w\). To evaluate, replace \(w\) with the given values, and compute the result.
To evaluate for \(w = 4\):
- Substitute \(w\) in the function: \(r(4) = 1.8^4\).
- Calculate the power: \(1.8 \times 1.8 \times 1.8 \times 1.8 = 10.4976\).
- Hence, \(r(4) = 10.4976\).For \(w = -0.5\):
- Substitute \(w\) in the function: \(r(-0.5) = 1.8^{-0.5}\).
- Negative and fractional exponent means we take the reciprocal and square root: \(\frac{1}{\sqrt{1.8}} \approx 0.745\).
- So, \(r(-0.5) \approx 0.745\).Through function evaluation, we see how changing the input \(w\) can significantly alter the output value of the function. This is especially true for exponential functions, where even small changes in the exponent can result in large changes in the outcome.

Rounding numbers is an essential skill in mathematics to simplify results while maintaining a degree of accuracy. It involves altering the number to a specified level of precision, usually to make it easier to read or use in further calculations.
Per the exercise instructions, we need to round our results to three decimal places:

• The result \(10.4976\) becomes \(10.498\) because the fourth digit (7) means we round up the third decimal place.
• The result \(0.745\) stays as it is since it already has three decimal places.

This standard practice of rounding helps in estimating and comparing numbers more manageably, especially in practical applications where exact precision might not be possible or necessary. Proper rounding can also prevent errors from accumulating in lengthy calculations.