Problem 9 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The given expression with positive exponents becomes \(a^{5} * \frac{1}{b^8}\).

Step 1: Identify Negative Exponents

In the expression, one term has negative exponents which is \(b^{-8}\)

Step 2: Use the Rule of Negative Exponents

The rule of exponents tells us that to convert a term with negative exponent to positive, we need to take its reciprocal. So, \(b^{-8}\) would become \(\frac{1}{b^8}\). It should be noted that \(a^{5}\) already has a positive exponent so it remains as it is.

Step 3: Rewrite the Expression

Now the expression could be rewritten as \(a^{5} * \frac{1}{b^8}\) when exponent rules have been applied.

Key Concepts

Negative Exponents

When dealing with algebraic expressions, you might encounter exponents that are negative. Negative exponents can be a bit tricky at first glance, but the concept is straightforward once you understand the basic idea.

A negative exponent indicates that the base should be moved to the opposite part of a fraction.

• If a base has a negative exponent in the numerator, you move it to the denominator and make the exponent positive.
• Conversely, if it has a negative exponent in the denominator, moving it to the numerator will make the exponent positive.

For instance, in the original expression, we had a negative exponent as seen in \(b^{-8}\). We can convert it to a positive exponent by rewriting it as the reciprocal: \(\frac{1}{b^8}\).

This transformation simplifies the process of handling and simplifying expressions, as working with positive exponents is typically more intuitive and straightforward.

Exponent Rules

Exponent rules are fundamental to simplifying expressions involving powers. These rules help manage the multiplication and division of expressions with the same base. They include a few key ideas:

• Product of Powers Rule: When you multiply two powers with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Negative Exponent Rule: Moving a base with a negative exponent involves using its reciprocal: \(a^{-n} = \frac{1}{a^n}\).
• Zero Exponent Rule: Any non-zero base raised to the exponent of zero is one: \(a^0 = 1\).

In our example, the primary rule applied was the negative exponent rule, which helped transform \(b^{-8}\) into a positive exponent form of \(\frac{1}{b^8}\).

These rules form the backbone of manipulating algebraic expressions and are essential for progressing in algebraic problems.

Simplifying Expressions

Simplifying expressions is an integral process in algebra. It involves reducing expressions to their simplest form while maintaining the same value. The steps to simplify expressions often include applying the rules of exponents and combining like terms.

In the case of our particular expression \(a^{5} b^{-8}\), the process was:

• Identify terms with negative exponents.
• Apply the negative exponent rule to modify terms with these exponents.
• Rewrite the expression in a more readable and simplified form.

By converting \(b^{-8}\) into \(\frac{1}{b^8}\) and leaving \(a^5\) as is, since it's already in its simplest form, we managed to rewrite the expression as \(a^5 \cdot \frac{1}{b^8}\).

Mastering the simplification of expressions is crucial for solving equations efficiently and accurately in algebra, making it easier to deal with complex calculations in the future.