Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. Round your result to three decimal places.\(1500\left(2^{-5 / 2}\right)\)

Step-by-Step Solution

Answer

After calculating and rounding off, the answer of the expression is approximately 53.033.

Step 1: Calculate the Exponent

Calculate the value of \(2^{-5/2}\) using a calculator. The negative exponent means you take the reciprocal of the number and the fractional exponent means you take the root. For the value of -5/2, this means the reciprocal of the square root of 2 to the power of 5.

Step 2: Multiply the Result by 1500

After getting the result of \(2^{-5/2}\), multiply this number by 1500. This gives the final result of the expression.

Step 3: Round the Result

The final step is to round the result to three decimal places. To do this, look at the fourth decimal place. If the number in the fourth decimal place is 5 or greater, round up the third decimal place. If it is less than 5, leave the third decimal place as it is.

Key Concepts

Exponentiation

Exponentiation is a fundamental operation in mathematics, much like addition or multiplication. It involves raising a base number to a certain power. The expression \( a^b \) means multiplying \( a \) by itself \( b \) times.

• For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).

When dealing with exponentiation, it's important to understand both the role of the base and the exponent. The base is the repeated factor and the exponent indicates how many times the base is used in the multiplication. Exponentiation can also be used to express very large or very small numbers efficiently, which is why it's so widely used in scientific notation for simplifying calculations. These concepts are essential as they lay the groundwork for more advanced topics in mathematics.

Negative Exponents

Negative exponents can be a bit tricky at first, but they're not too difficult once you understand the basic concept. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.

• For instance, \( 2^{-3} \) means \( \frac{1}{2^3} \) or \( \frac{1}{8} \).

In general, \( a^{-n} = \frac{1}{a^n} \). This reciprocal concept simplifies computations, especially when dealing with division in exponential equations. Remember, calculating with negative exponents involves inverting the base, which transforms what might appear to be a complex multiplication into an easier division problem. Mastery of this concept enables smoother navigation through algebraic manipulations and lays the foundation for understanding fractional and irrational exponents.

Rounding Numbers

Rounding numbers is a fundamental skill in math, particularly useful in ensuring that numbers are easier to work with or more understandable to interpret. When you round a number, you limit the digits while attempting to stay close to the original value.

• For instance, rounding 3.146 to one decimal place gives 3.1 because the digit in the second decimal place is less than 5.

The general rule is to check the digit one place beyond where you want to round to determine whether to "round up" or "round down." If this digit is 5 or greater, you round up; if it's less, you keep the number as is. Rounding becomes vital when presenting results to a set number of significant figures, often required in fields such as science and engineering to maintain precision without being overly complex.

Fractional Exponents

Fractional exponents represent both powers and roots. They provide a compact way of expressing roots, such as square roots or cube roots, using exponents. The numerator of the fractional exponent indicates the power, while the denominator indicates the root.

• For example, \( 2^{3/2} \) means the square root of \( 2^3 \), which is \( \sqrt{8} \).

A fractional exponent of \( n/m \) translates into \( \sqrt[m]{a^n} \). This dual operation can simplify the representation and computation of roots and powers in mathematical expressions. Understanding fractional exponents is crucial as it leads to greater fluency with radical expressions and operations, expanding the toolbox of methods available for problem-solving in both pure and applied mathematics.