Problem 12 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round your answer to the nearest ten thousandth. $$ 7^{-1} \cdot 7^{-3} $$

Step-by-Step Solution

Answer

The evaluated expression rounded to the nearest ten thousandth is 0.0002.

Step 1: Note the Properties of the Exponent

Recall the exponent rule that states when bases are the same and they are being multiplied, their exponents can be added. Thus, if you have \(a^n\) \(\cdot\) \(a^m\), it equals \(a^{n+m}\). This concept will be applied to the given problem.

Step 2: Apply the Exponents Rule

Applying the rule of exponents to the problem \(7^{-1} \cdot 7^{-3}\), the bases are the same (7) and they are being multiplied. Therefore, add the exponents together, which will give \(7^{-1-3}\) or \(7^{-4}\).

Step 3: Evaluate using a Calculator

Next, evaluate the expression \(7^{-4}\) using a calculator. Remember, reversing a positive exponent yields a reciprocal so \(7^{-4}\) is same as \(1/7^{4}\).

Step 4: Round to the Nearest Ten Thousandth

After evaluating the expression with the calculator, round the answer to the nearest ten thousandth as required by the problem.

Key Concepts

Negative Exponents

Negative exponents are an important concept in mathematics. They represent the reciprocal of a base raised to the corresponding positive exponent. This can initially seem confusing, but it simply flips the base. For example, when you see \(a^{-n}\), think of it as \(\frac{1}{a^n}\). This means that if you have \(7^{-1}\), it actually represents \(\frac{1}{7}\), and \(7^{-4}\) would be \(\frac{1}{2401}\) after calculating the fourth power of 7.

• Negative exponents move the base from the numerator to the denominator.
• They make very large numbers smaller and small numbers larger.
• Use the reciprocal method to simplify expressions involving negative exponents.

This visualization can help you tackle expressions like \(7^{-1} \cdot 7^{-3}\) by understanding the transition into its reciprocal form.

Multiplication of Exponents

When dealing with exponents of the same base, a shortcut can dramatically simplify your calculations: you can add the exponents together. Imagine working with a problem such as \(7^{-1} \cdot 7^{-3}\). Here, the base (7) is the same in both terms. So, according to the rule of multiplying exponents, you simply add the exponents: \(-1\) and \(-3\), which results in \(-4\). Thus, the expression simplifies to \(7^{-4}\).

• Add exponents directly when bases are the same.
• Simplifies complex exponent multiplication.
• Results in a new single exponent.

Realizing this property applies can transform a seemingly complicated process into an easy simplification task. Let it save you time and effort!

Rounding Numbers

Once you've calculated an expression, you might need to round the result to a specific decimal place. Rounding is a handy tool for simplifying numbers and making them easier to understand. When you round to the nearest ten thousandth, you focus on the fourth digit after the decimal point. Here’s how you do it:

• Identify the digit in the position you need to round to (ten thousandth).
• Look at the digit immediately to its right to decide whether to round up or keep it the same.
• If that digit is 5 or greater, round the earlier digit up.
• Keep the earlier digit the same if the next number is less than 5.

Rounding can remove unnecessary decimal places and help you work with more manageable figures, perfect when you need to present neat and concise results.