Problem 121 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(a^{6}\right)^{-2} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression with only positive exponents: \(\left(a^{6}\right)^{-2}\). Answer: \(\frac{1}{a^{12}}\)

Step 1: Use the power rule of exponents

We are given the expression \(\left(a^{6}\right)^{-2}\). By applying the power rule, we get: $$ a^{6 \times (-2)} $$

Step 2: Perform the multiplication

Now we multiply the exponents 6 and -2 to obtain the exponent for a: $$ a^{-12} $$

Step 3: Rewrite with positive exponent

To rewrite the expression with a positive exponent, apply the rule \(a^{-m} = \frac{1}{a^m}\) which states that a negative exponent in the numerator can be converted to a positive exponent in the denominator. So we have: $$ \frac{1}{a^{12}} $$ The final expression with only positive exponents is \(\frac{1}{a^{12}}\).

Key Concepts

Exponent Rules

Understanding the rules of exponents is essential for simplifying expressions and calculating powers efficiently. The exponent, also known as the power, tells us how many times a base is used as a factor. There are several rules to keep in mind:

• Product Rule: When multiplying two terms with the same base, add their exponents, expressed as \(a^m \times a^n = a^{m+n}\).
• Quotient Rule: For the division of terms with the same base, subtract the exponent of the denominator from the exponent of the numerator, shown as \(\frac{a^m}{a^n} = a^{m-n}\).
• Power Rule: When raising a power to another power, multiply the exponents together. This is expressed as \((a^m)^n = a^{mn}\), and it's the main focus of our example problem.

These rules help streamline calculations and simplify expressions effectively. They are particularly useful when dealing with complex algebraic operations.

Power Rule of Exponents

The power rule of exponents is a key concept in algebra. It states that when you raise an exponent to another exponent, you multiply the exponents together. This can be easily remembered as \((a^m)^n = a^{mn}\).Consider the expression \((a^6)^{-2}\):- Here, the base is \(a\), the first exponent is 6, and the second exponent is -2.- Applying the power rule, you multiply the exponents: \(6 \times (-2) = -12\).- As a result, the expression simplifies to \(a^{-12}\).This calculation helps us convert complex powers into simpler forms, making algebraic manipulations more manageable.

Negative Exponents

Negative exponents can be confusing, but they serve an important purpose in mathematics. A negative exponent indicates that the base should be taken as a reciprocal with a positive exponent.For instance, when we have \(a^{-m}\), it translates to \(\frac{1}{a^m}\). Let's see how this rule applies to our example:- The expression \(a^{-12}\) is in negative exponent form.- According to the rule, we convert this to \(\frac{1}{a^{12}}\).This conversion allows us to express any term with only positive exponents, which is often required in mathematical expressions and equations for clarity and simplicity.Understanding negative exponents simplifies the process of rewriting expressions and ensures they're presented correctly in solutions.