Problem 45 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ \left(4^{2}\right)^{-1} $$

Step-by-Step Solution

Answer

The rounded answer to the nearest ten thousandth is 0.0625.

Step 1: Simplify the expression using exponent rules

Simplify \( (4^{2})^{-1} \). Based on the rules of exponents, the exponent -1 means take the reciprocal of the inner expression. This gives us \( (4^{2})^{-1} = 1/(4^{2}) \). Now compute \( 4^{2} \) which equals 16.

Step 2: Calculate the reciprocal

Calculate \( 1/(4^{2}) = 1/16 \) using a calculator. This gives approximately 0.0625.

Step 3: Round to the nearest ten thousandth

Finally, round this value to the nearest ten thousandth. The result is 0.0625, which does not need any further rounding since it is already to the nearest ten thousandth.

Key Concepts

Understanding the Reciprocal of an Exponent

When you encounter an expression like \((a^n)^{-1}\), you're looking at an exponent's reciprocal. The reciprocal of a number is simply 1 divided by that number. Similarly, the reciprocal of an exponent is taking the inverse of the value it represents. For instance, \(a^n\) refers to multiplying \(a\) by itself \(n\) times. Conversely, \(a^{-n}\) involves division—specifically dividing 1 by \(a^n\).

In the expression \((4^2)^{-1}\), the exponent \(-1\) signals we should find the reciprocal of \(4^2\). By calculating \(4^2\), we get 16, so the reciprocal is \(1/16\). What's happening here is a fundamental exponent rule: applying a negative exponent effectively flips the base to its reciprocal form. This is an essential concept that can simplify many complex algebraic expressions.

Rounding Decimals to the Nearest Ten Thousandth

Precision is key in mathematics, and rounding decimals to a specific place value ensures uniformity and accuracy. To round a decimal to the nearest ten thousandth, you look at the fifth decimal place. If it is 5 or greater, you increase the fourth decimal place by one. If it's less than 5, you leave the fourth place as is.

In our case, the expression \(1/16\) gives us 0.0625, where the '5' is already in the fourth decimal place. Since there is no fifth decimal place to assess, we don't need to round up or down. When rounding, it's important to know the significance of each digit after the decimal. Each represents a fraction of a whole: tenths, hundredths, thousandths, ten thousandths, and so on. This level of precision is particularly important in fields such as finance, engineering, and science, where even minuscule discrepancies can lead to significant differences.

Evaluating Expressions with Exponent Rules

Evaluating expressions, especially those involving exponents, requires a solid understanding of exponent rules. When presented with an expression like \((4^2)^{-1}\), it's essential to systematically apply these rules to accurately compute the value.

First, knowing that an exponent signifies repeated multiplication helps comprehend why \(4^2\) equals 16. Next, understanding the purpose of a negative exponent—converting a base into its reciprocal—allows for the simplification of \((4^2)^{-1}\) into \(1/16\). Finally, using a calculator to perform the division ensures precision and simplifies the calculation process. The rules of exponents are not just arbitrary; they reflect profound properties of numbers and operations that maintain consistency throughout mathematics.