Problem 23 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ b^{-12} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $b^{-12}$ with only positive exponents. Answer: $\frac{1}{b^{12}}$

Step 1: Expression

Write the following expressions using only positive exponents. Assume all
variables are nonzero.
$$
b^{-12}
$$

Step 2: Exponent Rules

\(a^0=1\), \((a^m)^n=a^{mn}\), \(a^m a^n=a^{m+n}\).

Step 3: Result

Question: Rewrite the expression $b^{-12}$ with only positive exponents. Answer: $\frac{1}{b^{12}}$

Key Concepts

Negative Exponents

Negative exponents can be a bit tricky at first, but they follow a simple rule. Whenever you see a negative exponent, it means you are dealing with a reciprocal idea. For instance, when you have a term like \(b^{-12}\), this means you should take the reciprocal of the base \(b\) raised to the positive exponent. So, \(b^{-12}\) becomes \(\frac{1}{b^{12}}\). In essence, negative exponents flip the base to the other side of the fraction line:

• \(b^{-n} = \frac{1}{b^n}\)
• This applies to any base \(a\), so \(a^{-m} = \frac{1}{a^m}\)

Practice turning negative exponents into positive ones by remembering this reciprocal rule. The more you practice, the more natural it becomes!

Exponent Rules

Understanding exponent rules is essential in simplifying expressions in algebra. One of the most important rules when dealing with negative exponents is the Negative Exponent Rule:

• Negative Exponent Rule: \( a^{-m} = \frac{1}{a^m} \)

This tells us that a negative exponent represents the reciprocal of the base raised to the equivalent positive exponent.
Other helpful exponent rules include:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)

Recognizing and applying these rules makes it much easier to manipulate expressions and solve algebra problems effectively.

Algebra Expressions

Algebra expressions are mathematical phrases involving numbers, variables, and operators. They can vary in complexity from simple terms like \(x+2\) to more intricate ones with multiple variables and exponents. When you encounter exponents in algebraic expressions, applying the exponent rules to simplify or manipulate the expression is crucial.
For example, in an expression like \(b^{-12}\), using the negative exponent rule, we simplify it to \(\frac{1}{b^{12}}\) as shown. Simplification helps make expressions easier to work with and interpret.

• Always ensure to handle different terms using the correct algebra rules
• Stay mindful of operations like addition, subtraction, and multiplication involving variables
• Any expression with a negative exponent should be transformed into a positive exponent where possible

Understanding these fundamental concepts aids in effectively tackling more challenging algebraic problems and prepares you for more advanced mathematics.