Problem 42 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ 11^{-2} $$

Step-by-Step Solution

Answer

The value of the expression \(11^{-2}\), rounded to the nearest ten thousandth, is approximately 0.0083.

Step 1: Understand the instruction

The exponent -2 signifies that the base, 11, is to be squared and then reciprocated. That means \(11^{-2} = \frac{1}{11^2}\).

Step 2: Calculate the square of the base

Calculate the square of 11. So, \(11^2 = 121\).

Step 3: Reciprocal of the result

Reciprocate the obtained squared result. So, \(\frac{1}{11^2} = \frac{1}{121}\).

Step 4: Calculate the decimal equivalent

Using a calculator, find the decimal equivalent of \(\frac{1}{121}\), which equals to approximately 0.008264.

Step 5: Round off to the nearest ten thousandth

The result is then rounded off to the nearest ten thousandth, giving a final answer of 0.0083.

Key Concepts

Negative Exponent Calculation

Understanding negative exponents is a fundamental aspect of algebra that can initially confuse some students. To simplify, consider that a negative exponent indicates that you need to take the reciprocal of the base's positive exponent. For instance, when you have an expression like \( 11^{-2} \), it may look daunting at first, but all it means is that you should first calculate \( 11^2 \), and then take the reciprocal of that number.

When dealing with negative exponents, it's helpful to remember the basic rule: \( a^{-n} = \frac{1}{a^n} \) where 'a' is the base and 'n' is the positive exponent. Rather than thinking of the exponent as negative, you can simply flip the base to the bottom of a fraction and make the exponent positive. As shown in the example, \( 11^{-2} = \frac{1}{11^2} = \frac{1}{121} \).

Reciprocal of a Number

The reciprocal of a number is what you multiply that number by to get the product of 1. It's like asking, 'What can I multiply this by to get 1?' In mathematical terms, for any non-zero number 'x', its reciprocal is \( \frac{1}{x} \). If you have a fraction, such as \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \). Reciprocals are pivotal when working with negative exponents, as they're part of the rule for simplifying them.

The earlier exercise involved finding the reciprocal of \( 11^2 \), which is a squared number. The squared number was 121, and its reciprocal is \( \frac{1}{121} \). This transformation is crucial when dealing with expressions that have negative exponents.

Decimal Conversion

After understanding the concept of reciprocals, the next step is decimal conversion. This involves transforming a fraction into a decimal, which is often more understandable and more useful in real-world applications or further calculations. To convert a fraction like \( \frac{1}{121} \) into a decimal, you would divide the numerator by the denominator.

When you perform this division using a calculator, you get a long decimal number. In many cases, you'll need to round this number to make it more manageable. For example, \( \frac{1}{121} \) approximately equals 0.008264, yet when rounding to the nearest ten thousandth, it becomes 0.0083. Rounding is an essential skill as it makes numbers easier to work with and communicate to others, especially in scientific and engineering contexts where precision of a certain degree is required.