Problem 19 - College Algebra

Question

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. $$ 7^{-6} \cdot 7^{-3} $$

Step-by-Step Solution

Answer

Combine exponents: \( \frac{1}{7^9} \).

Step 1: Understanding the Problem

The problem requires combining the given expression into a single base using the properties of exponents. Here, we are multiplying two expressions with the same base 7.

Step 2: Identifying the Rule

Recall the property of exponents: when multiplying like bases, you add the exponents. The rule can be written as \( a^m \cdot a^n = a^{m+n} \).

Step 3: Applying the Rule

Apply the exponent multiplication rule to the expression: \( 7^{-6} \cdot 7^{-3} = 7^{-6 + (-3)} \).

Step 4: Simplifying the Exponent

Add the exponents: \( -6 + (-3) = -9 \). Therefore, the expression becomes \( 7^{-9} \).

Step 5: Rewriting with Positive Exponents

Rewrite the expression with a positive exponent by taking the reciprocal: \( 7^{-9} = \frac{1}{7^9} \).

Key Concepts

Multiplication of Exponents

Understanding the multiplication of exponents is crucial when working with expressions having the same base. When you have a base that repeats across multiple terms, like in the expression

• \(7^{-6} \cdot 7^{-3}\),

you can use the rule for multiplying exponents to simplify the expression. This rule states that when you multiply powers with the same base, you simply add their exponents together. This makes it easier because instead of multiplying large numbers, you just focus on the exponents.
Let's break it down: For any numbers with the same base \(a\),

• \(a^m \cdot a^n = a^{m+n}\).

So for our expression, you add the exponents:

• \(-6 + (-3)\)

This results in \(7^{-9}\). It simplifies your work by transforming two terms into one single term.

Always remember, though, this rule only works if the bases are the same! If they're different, the rule doesn't apply.

Negative Exponents

Negative exponents are a way of expressing numbers that are essentially smaller than 1. If you see a negative exponent, it actually indicates a fraction rather than a negative number. Here's how it works:

• \(a^{-n} = \frac{1}{a^n}\).

This says that a negative exponent \(-n\) instructs you to take the reciprocal (1 divided by) of the base raised to the positive \(n\). For our expression, we obtained \(7^{-9}\). This means:

• \(7^{-9} = \frac{1}{7^9}\).

This form is more user-friendly, especially when specific scenarios require positive exponents only. Negative exponents are especially useful because they let you denote very small numbers in decimal form succinctly. Remember, they're about reciprocals, not negativity.

Rewriting Expressions with Positive Exponents

Rewriting expressions to only include positive exponents is often necessary in math. Positive exponents are generally more straightforward to understand and work with, whether it's for solving equations or performing other operations. Here's the step you follow:

• If you have an expression like \(7^{-9}\), you convert it into positive exponents by taking it to the denominator: \(\frac{1}{7^9}\).

This conversion is made because, in everyday calculations and final answers, positive exponents present a cleaner look and are easier for others to interpret.
Not only does this make computations easier in many cases, but it also aligns with general practices in scientific notation and significant number presentation in fields like science and engineering. Rewriting to positive exponents doesn't change the value, but merely presents it in another form.