Problem 31 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (r+3)^{-8} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression with a negative exponent as an expression with only positive exponents: $$ (r+3)^{-8} $$ Answer: $$ \frac{1}{(r+3)^8} $$

Step 1: Identify the expression with the negative exponent

Given expression is: $$ (r+3)^{-8} $$ In this expression, \((r+3)\) has a negative exponent of \(-8\).

Step 2: Apply the property of negative exponents

According to the property of negative exponents, we can rewrite the expression as: $$ (r+3)^{-8} = \frac{1}{(r+3)^8} $$

Step 3: Write the final expression with positive exponents

Now that we've applied the property of negative exponents, the final expression with only positive exponents is: $$ \frac{1}{(r+3)^8} $$

Key Concepts

Positive Exponents

Understanding positive exponents is foundational to manipulating algebraic expressions effectively. When a number or variable is raised to a positive exponent, it signifies repeated multiplication.

For instance, the expression \(2^3\) indicates that the number 2 is multiplied by itself 3 times, leading to \(2 \times 2 \times 2\) which equals 8. This process is the same for algebraic terms, such as \(x^2\), which would equate to \(x \times x\).

When transitioning from negative to positive exponents, it's crucial to recognize that positive exponents reflect direct multiplication, while negative exponents represent the reciprocal process. An expression like \(x^{-3}\) is essentially \(\frac{1}{x^3}\), where the '3' has been converted from a negative to a positive exponent by placing the term in the denominator.

Exponent Properties

Exponent properties, or laws of exponents, are rules that govern the operations involving exponents. These rules make it possible to simplify and manipulate expressions without altering their values.

Some of the fundamental exponent properties include the following:

• Product of Powers: To multiply powers with the same base, add their exponents. For example, \(x^a \times x^b = x^{a+b}\).
• Quotient of Powers: To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator, as in \(\frac{x^a}{x^b} = x^{a-b}\).
• Power of a Power: To raise an exponent to another exponent, multiply the exponents. So, \(\left(x^a\right)^b = x^{a \cdot b}\).
• Negative Exponent: A negative exponent signifies the reciprocal of the base raised to the positive counterpart of the exponent, meaning \(x^{-a} = \frac{1}{x^a}\).

In the provided exercise, the negative exponent rule is applied by transforming \(r+3)^{-8}\) into \(\frac{1}{(r+3)^8}\), which uses the concept that \(x^{-n} = \frac{1}{x^n}\) to achieve an equivalent expression with only positive exponents.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols such as plus, minus, multiplication, and division signs. They can range from simple, like \(3x + 2\), to more complex expressions involving exponents, parentheses, and various levels of nested operations.

Simplifying algebraic expressions often involves combining like terms or applying exponent properties. In this context, 'like terms' are terms that have the same variables raised to the same power. They're the building blocks that you can manipulate to simplify expressions without changing the equations or inequalities they're part of.

A key part of working with algebraic expressions is understanding how to rewrite them to clarify their components or convert them into a preferable format, as with converting negative exponents to positive ones. A clear grasp of the various properties of exponents is essential in these manipulations, as they allow you to reconfigure the expressions according to the needs of the problem you're solving.