Problem 44 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ 9^{-4} \cdot 9^{2} $$

Step-by-Step Solution

Answer

The result of 9^{-4} * 9^{2} rounding to the nearest ten thousandth is 0.0123.

Step 1: Use Exponent Rules

The expression can be simplified using the exponent rule a^m/a^n = a^(m-n), resulting in: 9^(-4+2) = 9^(-2)

Step 2: Calculate

Next, calculate 9^(-2). \\In your calculator, the negative exponent means that 9^(-2) equals 1/9^2 = 1/81.

Step 3: Conversion to Decimal and Rounding

The result 1/81 is converted to decimal form by dividing, and then rounded to the nearest ten thousandth. The rounded value is equal to 0.0123.

Key Concepts

Negative Exponents

Negative exponents can be a bit tricky at first glance, but understanding them is quite simple once you get the hang of the rules. A negative exponent indicates that you are dealing with a reciprocal. For example, when you see something like \( a^{-n} \), it means \( \frac{1}{a^n} \). So instead of multiplying \( a \) by itself \( n \) times, you're dividing 1 by \( a \) multiplied by itself \( n \) times. Think of it as flipping the fraction.

This concept is quite useful in simplifying expressions, especially when combined with other exponent rules like multiplying powers with the same base. As seen in the original exercise, we applied the rule of adding exponents when multiplying terms with the same base: \( 9^{-4} \cdot 9^{2} = 9^{-4+2} = 9^{-2} \). Thus, calculating an expression with a negative exponent becomes a matter of using the reciprocal of its positive counterpart. So, \( 9^{-2} \) is the same as \( \frac{1}{9^2} = \frac{1}{81} \).

Breaking down a problem into these steps makes handling negative exponents much more manageable.

Rounding Decimals

When you need to round decimals, you often need to follow specific rules to ensure accuracy in your calculations. Rounding decimals helps in approximating values to make them easier to work with, especially when results have long decimal expansions.

In the given exercise, after solving \( 9^{-2} \) which equals \( \frac{1}{81} \) or approximately 0.012345679, rounding is the next step. You round this result to the nearest ten thousandth, which is the fourth digit after the decimal point.

Here's how you round decimals effectively:

• Look at the digit in the place immediately right of the "target" digit (in our case, the fifth digit).
• If the digit to the right is 5 or greater, increase the "target" digit by one. If it's less than 5, keep the "target" digit the same.
• In our example, 0.012345679 rounds to 0.0123.

This process is essential for maintaining precision in calculations while making numbers more straightforward to interpret.

Calculator Usage

A calculator is a powerful tool that can help you solve complex calculations quickly and accurately. Understanding how to use one efficiently can enhance your mathematical skills significantly.

For our exercise, you first need to handle negative exponents by using the calculator's exponentiation feature. Most calculators use the caret symbol (\(^\wedge\)) or an "EXP" button to signify exponents.

Here’s a brief guide to using a calculator for negative exponents:

• Input the base number.
• Access the exponent mode, usually via an "EXP" or "^" button.
• Enter the negative exponent (e.g., "-2" for \(9^{-2}\)).

Once you obtain the decimal equivalent, you often want to round it as required by your problem, in this case to the nearest ten thousandth.
Calculators are also great for conversion tasks, as they quickly toggle large or small numbers into rounded decimals ready for your analysis. Mastering these basic operations will make handling math tasks more straightforward.