Problem 11 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ 5^{-4} $$

Step-by-Step Solution

Answer

The final answer to \(5^{-4}\) rounded to the nearest ten-thousandth place is 0.0016.

Step 1: Understand the Concept - Negative Exponents

A negative exponent indicates that we have to take the reciprocal (one over) of the base. So \(5^{-4}\) should be understood as \(\frac{1}{5^4}\).

Step 2: Evaluate Expression

The first step is to calculate the value of \(5^4\) which results in 625. So, our expression now becomes \(\frac{1}{625}\).

Step 3: Calculate Reciprocal

Next, we calculate the reciprocal of 625 which results in a decimal number 0.0016.

Step 4: Rounding

Finally, we round our answer to the nearest ten-thousandth place. In this case, it doesn't change as the value is already at the ten-thousandth place.

Key Concepts

Evaluate Expressions Using Calculators

When faced with complex mathematical expressions, especially those involving exponents, a calculator can be an invaluable tool. To evaluate an expression like \(5^{-4}\), we must interpret the negative exponent and input the calculation correctly into the calculator. This process involves understanding that a negative exponent represents the reciprocal of the base raised to the positive exponent. Thus, \(5^{-4}\) becomes \(\frac{1}{5^{4}}\).

Using a calculator, enter the base (5) followed by the exponent (4), ensuring to use the correct exponentiation key, often labeled as 'x^y' or '^'. Once you have the value of \(5^{4}\), which is 625, you can then find its reciprocal. This is typically done by pressing the '1/x' button or entering '1 divided by 625' to get \(\frac{1}{625}\) which is 0.0016. The calculator does the conversion quickly and accurately, allowing students to check their understanding of the concept of negative exponents.

Rounding Decimal Numbers

Rounding decimals is a fundamental concept in mathematics, designed to simplify numbers to make them easier to work with. The process involves approximating a number to a specific decimal place. When rounding the answer of \(5^{-4}\) evaluated as 0.0016, we must consider the decimal place to which we're rounding. If we're rounding to the nearest ten-thousandth, we look at the fifth decimal place, if it exists. Since 0.0016 is exactly at the ten-thousandth place, no rounding is needed here.

However, if we had a number like 0.00165 and needed to round to the nearest ten-thousandth, we would look at the digit in the fifth place (5 in this case) and round up the fourth decimal place accordingly, resulting in 0.0017. It's also important to remember to include zeros to maintain the rounding precision, as in rounding to 0.0020 if necessary.

Reciprocal of a Number

The reciprocal of a number is essentially one divided by that number. It is also known as the multiplicative inverse, since when a number is multiplied by its reciprocal, the result is one. For instance, the reciprocal of 2 is \(\frac{1}{2}\), and when you multiply 2 by \(\frac{1}{2}\), you indeed get 1.

In the context of negative exponents, like \(5^{-4}\), we look at the base (5) and find its reciprocal as part of the process. For \(5^{-4}\), the reciprocal is \(\frac{1}{5^4}\) or \(\frac{1}{625}\), which is 0.0016 after evaluation with a calculator. Understanding reciprocals is crucial when dealing with negative exponents and in many other mathematical operations involving division and fractions.