Problem 38 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ (1 \cdot 7)^{-3} $$

Step-by-Step Solution

Answer

The evaluated expression is \(\frac{1}{343}\).

Step 1: Simplify the expression in the parentheses

First, simplify what's inside the parentheses. The expression \(1 \cdot 7\) simplifies to \(7\). So the new expression is \(7^{-3}\).

Step 2: Apply the property of negative exponents

A negative exponent tells us to take the reciprocal of the base raised to the absolute value of that exponent. Therefore, \(7^{-3}\) is equivalent to \(\frac{1}{7^3}\).

Step 3: Evaluate the denominator

The expression in the denominator, \(7^3\), can be simplified. The expression means 7 multiplied by itself twice, that is, \(7 \cdot 7 \cdot 7 = 343\). So the final simplified expression is \(\frac{1}{343}\).

Key Concepts

Negative Exponents

Negative exponents might seem scary at first, but they are simply a way of expressing fractions or decimals in a different form. When you see a negative exponent, it means you will need to find the reciprocal of the base number and then apply the positive exponent. For example, if you encounter an expression like \(7^{-3}\), think of the negative exponent as instructions telling you to "flip" the base number to its reciprocal.

Here's the general rule: If \(a\) is a non-zero number, then \(a^{-n} = \frac{1}{a^n}\). So in this exercise, the expression \(7^{-3}\) can be rewritten as \(\frac{1}{7^3}\). It's that simple!

Always remember:

• A negative exponent suggests that the base should be in the denominator.
• The negative exponent becomes a positive when the base is flipped into its reciprocal form.

Understanding the concept of negative exponents will make many algebraic expressions much easier to handle.

Simplifying Expressions

Simplifying expressions is all about making them look as simple as possible while retaining the same value. Imagine you're cutting away the clutter to see the core idea clearly. It's like turning a complex puzzle into a straightforward picture.

For instance, let's go back to our problem. We started with \((1 \cdot 7)^{-3}\). By simplifying inside the parentheses, our expression became \(7^{-3}\). This process doesn't change the value but helps us focus only on the base and its exponent. In other words:

• Simplify inside brackets first.
• Look for any multiplication or division inside the brackets.
• Apply any math rules you know (like the rules for exponents).

By following these steps, you can simplify expressions and make them easier and quicker to evaluate.

Reciprocal

The word 'reciprocal' often pops up when dealing with negative exponents. In simple terms, the reciprocal of a number is what you get when you divide 1 by that number. For instance, the reciprocal of 5 is \(\frac{1}{5}\). When you multiply a number by its reciprocal, the result is 1.

Now, let’s consider how reciprocals are used with negative exponents. When you see a negative exponent, it implies you need to take the reciprocal of the base. For example, \(7^{-3}\) means taking the reciprocal of 7 and applying the exponent: \(\frac{1}{7^3}\).

Key points to remember:

• Reciprocal means 1 divided by the number.
• A negative exponent transforms the expression into its reciprocal form.
• The reciprocal helps in simplifying expressions involving division and negative powers.

Recognizing how the concepts of negative exponents and reciprocals are intertwined makes evaluating expressions less daunting.