Problem 16 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ a^{5} b^{-8} $$

Step-by-Step Solution

Answer

The expression \(a^{5} b^{-8}\) with positive exponents is written as \(a^{5}\cdot\frac{1}{b^8}\).

Step 1: Identify the negative exponent

The negative exponent in this expression is '-8', which is the exponent of \(b\).

Step 2: Apply the property of negative exponents

According to the property of negative exponents, \(a^{-n} = \frac{1}{a^n}\) where a ≠ 0. Therefore, \(b^{-8}\) can be rewritten as \(\frac{1}{b^8}\).

Step 3: Rewrite the whole expression

Now, replace \(b^{-8}\) with \(\frac{1}{b^8}\) in the original expression. This gives us the final expression as \(a^{5}\cdot\frac{1}{b^8}\).

Key Concepts

Understanding Exponent Rules

When dealing with algebraic expressions, understanding exponent rules is crucial for simplifying expressions. Exponential notation is a way to show repeated multiplication. For example, to say that we are multiplying the base, let's call it 'a', by itself 5 times, we write it as a^5.

Dealing with negative exponents might seem tricky at first, but the rule is actually quite straightforward. A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. In other words, a^-n = 1/(a^n) for any nonzero number 'a' and positive integer 'n'. Remember, anything to the zero power, like a^0, is always 1.

• Positive exponent: a^n implies multiplying 'a' by itself 'n' times.
• Negative exponent: a^-n shows the reciprocal of the base raised to a positive power.
• Zero exponent: a^0 is always equal to 1.

These basic rules help us transform and simplify expressions to make them easier to work with.

Algebraic Expressions and Their Components

An algebraic expression is a mathematical phrase that can include numbers, variables (like a and b), and operation symbols like +, -, *, and /. Important components of these expressions are the coefficients (numerical part of a term with a variable), constants (terms with numbers only), and variables (symbols that represent unknown numbers).

Consider the expression used in the example, a^5 b^{-8}. Here, a and b are variables, and the power 5 and -8 are their respective exponents. No explicit coefficient is visible, which means the coefficient is 1.

By understanding the parts and purposes of algebraic expressions, it becomes easier to apply the rules of exponents and simplify the expression into a more digestible form.

Simplifying Expressions with Negative Exponents

The process of simplifying expressions, particularly those involving negative exponents, is about making the expression as straightforward as possible. For the expression a^5 b^{-8}, simplification means following the steps laid out in the solution. First, identify the negative exponent (which is -8), then apply the rule for negative exponents to rewrite b^{-8} as 1/b^8.

In the final step, you consolidate the expression to its simplest form without negative exponents: a^5 * 1/b^8, or simply a^5/b^8 when written in a single fraction form.

• Identify negative exponents.
• Apply the reciprocal rule for negative exponents.
• Rewrite the expression in a simplified form.

Following these steps helps ensure clarity and accuracy in algebraic work, making it easier for students to manage and solve complex problems.