Problem 94 - Precalculus: Functions and Graphs

Question

Approximate the real-number expression to four decimal places. (a) \((2.6-1.9)^{-2}\) (b) \(5^{\sqrt{7}}\)

Step-by-Step Solution

Answer

(a) 2.0408 (b) 160.5550

Step 1: Simplify the expression inside the parentheses for (a)

Calculate the difference inside the parentheses: \((2.6 - 1.9) = 0.7\).

Step 2: Apply exponent for (a)

Raise the result from Step 1 to the power of -2:\((0.7)^{-2} = \frac{1}{(0.7)^2}\).Calculate \((0.7)^2 = 0.49\), so the expression becomes \(\frac{1}{0.49}\).

Step 3: Calculate the reciprocal for (a)

Compute \(\frac{1}{0.49}\):This results in approximately 2.0408.

Step 4: Use a calculator for (b)

Calculate \(5^{\sqrt{7}}\) using a calculator:First, calculate \(\sqrt{7} \approx 2.6458\).Then raise 5 to the power of 2.6458.

Step 5: Approximate result for (b)

Using the result from Step 4:\(5^{2.6458} \approx 160.5550\).

Key Concepts

Real Numbers

Real numbers are the building blocks of mathematics that you likely encounter daily without even realizing it. Anything on the number line, from negative numbers, zero, to positive numbers and fractions, is a real number.
Real numbers include:

• Whole numbers: like 0, 1, 2, and 3.
• Fractions: such as 1/2 and 3/4.
• Decimals: like 2.5 or 0.75.
• Square roots or irrational numbers: numbers that cannot be exactly expressed as fractions, such as \(\sqrt{2}\), which is approximately 1.414.

Understanding real numbers is crucial because they allow us to measure, calculate, and approximate the unknown using known numerical systems. They form the foundation when approaching topics like exponents and square roots, which often involve manipulations and approximations involving real numbers.

Exponents

An exponent signifies how many times a number, known as the base, is multiplied by itself. This mathematical operation can significantly change the size of a number. If you see a number like \(3^4\), this means 3 is multiplied by itself 4 times, resulting in 81.
Exponents are essential in various calculations:

• Positive exponents (\(a^n\)) indicate regular multiplication.
• Negative exponents (\(a^{-n}\)) represent the reciprocal of multiplying that base to the positive exponent: \(a^{-n} = \frac{1}{a^n}\).
• Fractional exponents relate to roots: \(a^{1/n}\) equals the nth root of a.

In the exercise provided, part (a) deals with a negative exponent (\((0.7)^{-2}\)), which means taking the reciprocal of \(0.7\) squared. Understanding these properties of exponents is pivotal for exact and approximate calculations in algebra and beyond.

Square Roots

Square roots answer the question: "What number, when multiplied by itself, gives me this original number?" For the number 9, its square root is 3 because \(3 \times 3 = 9\).
Some important points about square roots include:

• The square root of a perfect square (like 4, 9, 16) results in whole numbers.
• Non-perfect squares (like 2, 3, or 7) result in irrational numbers, which are non-terminating, non-repeating decimals.
• Square roots can be approximated to simplify calculations in real-life applications.

In the provided exercise, part (b) uses the square root of 7, calculated approximately as 2.6458. This approximation shows how even complex expressions involving roots can be handled using numerical approximation, allowing calculations with irrational numbers to become manageable and helpful in decision-making processes.