Problem 32 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (a-1)^{-12} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $(a-1)^{-12}$ using positive exponents. Answer: The rewritten expression with a positive exponent is $\frac{1}{(a-1)^{12}}$.

Step 1: Identify the base and exponent

In the given expression $$(a-1)^{-12}$$, the base is \((a-1)\) and the exponent is \(-12\).

Step 2: Rewrite the expression using the rule for negative exponents

According to the rule, $$a^{-n} = \frac{1}{a^n}$$. Therefore, we can rewrite the expression as follows: $$ (a-1)^{-12} = \frac{1}{(a-1)^{12}} $$ The expression is now written with a positive exponent.

Key Concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication of a number or expression by itself. When we have an expression like \( x^n \), the number \( x \) is being multiplied by itself \( n \) times. For example, \( 2^3 \) means \( 2 \times 2 \times 2 \), which equals 8. Exponents follow specific rules that help in simplifying calculations:

• Product of Powers: When multiplying two powers with the same base, add their exponents. For example, \( a^m \times a^n = a^{m+n} \).


• Power of a Power: When raising a power to another power, multiply the exponents. For instance, \( (a^m)^n = a^{m \times n} \).


• Quotient of Powers: When dividing two powers with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

Understanding these rules is crucial as they form the basis of working with exponents in algebra.

Base and Exponent Relationship

In mathematics, expressions with exponents involve two main parts: the base and the exponent. The **base** is the number or expression that is being multiplied. In the example \( (a-1)^{-12} \), the base is \( (a-1) \). It's essentially the foundation of the power you're dealing with. The **exponent** indicates how many times the base is used as a factor. It is often represented as a small number located at the upper right of the base. In our example, the exponent is \(-12\). This tells us how many times to divide by the base when the exponent is negative, or how many times to multiply the base when positive. When dealing with expressions, recognizing which part is the base and which is the exponent is essential. This helps in easily applying rules for simplification and transformation of expressions, such as converting negative exponents to positive ones.

The Concept of Positive Exponents

Positive exponents are perhaps the simplest form of exponents. They tell us how many times to multiply the base by itself. For an expression \( a^n \), if \( n \) is positive, it means you perform the multiplication \( n \) times. For instance, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 = 81 \).Positive exponents are straightforward because there is no inversion or division involved, unlike negative exponents. The main rule to remember is that as soon as the exponent becomes positive, the arithmetic becomes just repeated multiplication.Why are positive exponents favorable?

• They simplify expressions, making them more intuitive to handle.


• They help in finding areas and volumes in geometric calculations, where direct multiplication is involved.


• In calculus, dealing with positive exponents often simplifies differentiation and integration.

In algebra, ensuring expressions are in positive exponent form whenever possible makes computations easier and helps in avoiding common errors, especially in calculations involving multiple steps.