Problem 70 - College Algebra with Modeling and Visualization

Question

Evaluate the expression by band. Approximate the answer to the nearest hundredth when appropriate. $$ 9^{-3 / 2} $$

Step-by-Step Solution

Answer

The result is approximately 0.037.

Step 1: Understand the Exponent

The expression to evaluate is \( 9^{-3/2} \). The exponent \(-3/2\) is a fraction which means we need to find the square root of \(9\), raise it to the third power, and then take the reciprocal since the exponent is negative.

Step 2: Square Root

Calculate the square root of \(9\). Since \(9\) is a perfect square, the square roots are \(3\) and \(-3\). We'll use \(3\) because we're interested in positive real numbers here. So, \( \sqrt{9} = 3 \).

Step 3: Cube the Result

Raise the result from Step 2 to the power of \(3\). Thus, \(3^3 = 27\).

Step 4: Take the Reciprocal

Because the original exponent is negative, take the reciprocal of the result from Step 3. The reciprocal of \(27\) is \( \frac{1}{27} \).

Step 5: Approximate to the Nearest Hundredth

Convert \( \frac{1}{27} \) to a decimal by dividing 1 by 27. This gives approximately \(0.037037...\). To the nearest hundredth, it is \(0.037\).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra used to express repeated multiplication. When you see a number like \(9^{-3/2}\), it's telling you to raise the base, which is \(9\), to the power of \(-3/2\). This might look complicated at first, but it simply means performing several operations:

• Fractional Exponent: The exponent \(-3/2\) is negative, indicating a reciprocal, and a fraction implies both a root and a power.
• Components: The \(2\) in the denominator suggests taking a square root, while the \(3\) in the numerator involves cubing the number.

To simplify, you can take the operations step-by-step, which not only clarifies the concept but makes managing the calculations easier. For example, we first took the square root of \(9\), then cubed the result, and finally took the reciprocal because of the negative sign in the exponent.

Square Roots

The square root is the number which when multiplied by itself yields the original number. In mathematical terms, if \( x^2 = 9 \), then \( x = \sqrt{9} \). Calculating square roots is crucial because it simplifies expressions before further operations, like raising to powers or taking reciprocals.When dealing with \(9\), a straightforward case since \(9\) is a perfect square, its square roots are \(3\) and \(-3\). However, we typically focus on the positive value because it represents the principal square root in many contexts. This simplification is particularly useful when dealing with further operations like exponents or equations.

Reciprocals

Reciprocals are foundational in mathematics, especially when working with negative exponents. The reciprocal of a number is essentially \(1\) divided by that number. For example, the reciprocal of \(27\) is \(\frac{1}{27}\).Negative exponents signify that we need to take the reciprocal of the base raised to the positive exponent. So for \(9^{-3/2}\), after calculating \(3^3 = 27\), we find its reciprocal which is \(\frac{1}{27}\). By finding the reciprocal, negative exponents help transform potentially large results into smaller, simpler decimals, providing a compact way to represent the inverse of multiplication.