Problem 79 - Algebra A Combined Function

Question

Simplify, if possible. $$ a \cdot b^{3} \cdot a^{2} \cdot b^{7} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^3 \cdot b^{10} \).

Step 1: Group Similar Terms

In the expression \( a \cdot b^{3} \cdot a^{2} \cdot b^{7} \), identify and group the similar variables together. We have two \( a \) terms: \( a \) and \( a^2 \), and two \( b \) terms: \( b^3 \) and \( b^7 \). Thus, the expression can be rearranged as \( (a \cdot a^2) \cdot (b^3 \cdot b^7) \).

Step 2: Apply the Product of Powers Property

The Product of Powers Property states that \( x^m \cdot x^n = x^{m+n} \). Apply this property to the grouped terms. For the \( a \) terms, \( a \cdot a^2 = a^{1+2} = a^3 \). For the \( b \) terms, \( b^3 \cdot b^7 = b^{3+7} = b^{10} \).

Step 3: Combine Simplified Terms

Combine the simplified powers of \( a \) and \( b \) from Step 2. The entire expression simplifies to \( a^3 \cdot b^{10} \).

Key Concepts

Product of Powers Property

The Product of Powers Property is a fundamental rule in algebra that helps simplify expressions that involve exponents. It states that when you multiply two expressions with the same base, you can add the exponents. This is because repeated multiplication can be combined into a single power.
For instance, if you have two terms like \( x^m \times x^n \), you can simplify it to \( x^{m+n} \). This simplification occurs because you are essentially adding the number of times \( x \) is used in multiplication.

• Example: \( x^2 \times x^3 = x^{2+3} = x^5 \).
• This property only applies to terms with the same base. This means \( x^m \times y^m \) cannot be simplified using this rule.

This property is particularly useful in simplifying algebraic expressions, as seen in the given exercise. Grouping terms with the same bases and using the Product of Powers Property can greatly reduce the complexity of an expression, making it easier to work with.

Exponents

Exponents are a way to express repeated multiplication of the same number. An expression like \( a^3 \) means that you multiply \( a \) by itself three times: \( a \times a \times a \).
Exponents make it easier to write large numbers or repeated multiplications succinctly. They are also essential in algebra for representing polynomial expressions and for simplifying calculations.

• The base of an exponent refers to the number being multiplied.
• The exponent itself indicates how many times the base is used as a factor.

Understanding how to manipulate exponents, such as multiplying and dividing them, is key in simplifying and solving algebraic expressions. When you see an expression like \( b^3 \times b^7 \), knowing that you add the exponents to get \( b^{10} \) helps in simplifying algebraic expressions effectively.

Algebraic Expression Simplified

Simplifying an algebraic expression involves reducing it to its simplest form. This often means eliminating unnecessary parts, combining like terms, and applying algebraic properties such as the Product of Powers Property.
When you simplify an expression, you make it easier to read and use. For example, the expression \( a \cdot b^3 \cdot a^2 \cdot b^7 \) can be simplified by first grouping similar terms. Here, you group \( a \) terms and \( b \) terms separately and then apply the exponent rules.

• Step 1: Group similar bases to handle them separately.
• Step 2: Apply properties like Product of Powers to simplify within each group.
• Step 3: Combine these simplified components into a single neat term.

The resulting expression, \( a^3 \cdot b^{10} \), is the simplest form, offering clarity and a better understanding of the relationship between variable terms. This simplification process is crucial for solving equations and makes further algebraic manipulations straightforward.