Problem 64 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 4 a^{2} b^{2} a^{-5} b^{-2} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(4a^{2}a^{-5}b^{2}b^{-2}\) using only positive exponents. Answer: \(\frac{4}{a^{3}}\)

Step 1: Combine like terms with the same base

The first step is to simplify the expression by considering the terms with the same base, i.e., \(a\) and \(b\). As we know, when multiplying terms with the same base, we can add their exponents. So the expression can be rewritten as: $$ (4 a^{2} a^{-5}) (b^{2} b^{-2}) $$ Step 2: Working on 'a'

Step 2: Apply exponent properties on 'a' terms

Now, let's simplify the expression we got above by applying the properties of exponents on 'a' terms: $$ a^{2} a^{-5} = a^{(2 - 5)} = a^{-3} $$ Step 3: Working on 'b'

Step 3: Apply exponent properties on 'b' terms

Now, let's simplify the expression we got by applying the properties of exponents on 'b' terms: $$ b^{2} b^{-2} = b^{(2 - 2)} = b^{0} $$ As we know, any non-zero number raised to the power of 0 is equal to 1. Therefore, \(b^{0}=1\). Step 4: Final Expression

Step 4: Assemble the final expression using only positive exponents

Now, let's write the final expression using only positive exponents by observing the results obtained above: $$ 4 a^{-3} b^{0} = 4 \cdot \frac{1}{a^{3}} \cdot 1 $$ Hence, the given expression can be rewritten as: $$ \frac{4}{a^{3}} $$

Key Concepts

Simplifying Algebraic Expressions

Simplifying expressions may sound complex at first, but it's all about making equations easier and more readable. When dealing with algebraic expressions that involve exponents, such as \(4a^2 b^2 a^{-5} b^{-2}\), our goal is to reduce the expression to its simplest form.

To simplify expressions like these:

• Identify similar terms: Group terms with the same base. For example, identify all terms containing \(a\) and \(b\).
• Apply properties of exponents: Handle the exponents correctly using rules like \(a^m \cdot a^n = a^{m+n}\).
• Rewrite using positive exponents: Convert negative exponents to positive by using reciprocal properties. For instance, \(a^{-n} = \frac{1}{a^n}\).

This approach allows you to streamline the expression while maintaining its value and integrity.

Understanding Properties of Exponents

The properties of exponents are essential tools in algebra that help us manage expressions involving powers. These properties guide us in simplifying expressions, ensuring we handle them correctly.

Here are the key properties you'll often use:

• Product of Powers: When multiplying like bases, add their exponents. \(a^m \cdot a^n = a^{m+n}\).
• Zero Exponent: Any base (except zero) raised to the power of zero is one. \(a^0 = 1\).
• Negative Exponent: Indicates division or reciprocation. \(a^{-n} = \frac{1}{a^n}\).

By mastering these properties, you can transform complex algebraic expressions using systematic and straightforward steps, just like converting \(4a^2 b^2 a^{-5} b^{-2}\) into a more manageable form.

Exploring Algebraic Expressions

Algebraic expressions form the backbone of algebra and are composed of numbers, variables, and operators like addition or multiplication. These expressions can vary from simple monomials to complex polynomials.

An algebraic expression includes:

• Constants: Numerical values (e.g., 4).
• Variables: Symbols that represent numbers (e.g., \(a\) and \(b\)).
• Exponents: Values that indicate the power to which a base is raised (e.g., \(a^2\)).

Understanding algebraic expressions involves recognizing how components interact. This knowledge is critical when simplifying expressions, as it requires unraveling and recombining different parts according to algebraic rules.

For instance, in \(4a^2 b^2 a^{-5} b^{-2}\), knowing each part's role helps us simplify it to \(\frac{4}{a^3}\). Mastering these concepts allows anyone to engage meaningfully with algebraic problems.