Question

\(\operatorname{Can} a^{3} \cdot b^{4}\) be simplified? Explain your answer.

Short answer

\(\operatorname{No,} a^{3} \cdot b^{4}\) cannot be simplified further. This is because the expression does not contain like terms or factors that can be combined according to the algebraic rules.

Step-by-step solution

Identify if the terms share anything in common

Here we see two terms: \(a^{3}\) and \(b^{4}\), raised to different powers. There is no common base nor common exponent that will allow for either the powers or the terms themselves to be combined under basic algebra rules.

Assess potential for simplification

In general, you can only simplify expressions that have like terms or factors. Like terms are terms that contain the same variables raised to the same power. Here, we don't have like terms or common factors since the variables are different and the powers are too.

Final Evaluation

After checking for like terms and common factors and finding none, the conclusion is that \(a^{3} \cdot b^{4}\) cannot be simplified further. This is because there are different bases (a and b) raised to different powers (3 and 4), and no algebraic rule allows these to be combined.

Key concepts

Like Terms in Algebra

When working with algebraic expressions, 'like terms' refer to terms that have exactly the same variable parts raised to the same powers. The coefficients of these terms can differ. Like terms are an important concept because they're the only terms that can be combined together through addition or subtraction.

For instance, if you have the expression \(2x^2 + 3x^2\), you can combine these like terms to get \(5x^2\). Notice that the variable part \(x^2\) is the same in both terms; hence, they are considered 'like' and can be simplified. In contrast, \(2x^2 + 3x\) cannot be simplified through addition or subtraction because the variable parts \(x^2\) and \(x\) are not the same.

In the exercise \(a^{3} \cdot b^{4}\), we cannot combine \(a^{3}\) and \(b^{4}\) since they do not have the same variable base, making them 'unlike terms'. Therefore, recognizing like terms is the first step towards simplifying complex algebraic expressions.

Exponents and Powers

Powers and exponents in algebra refer to the compact representation of repeated multiplication of the same number. The number that is being multiplied is called the base, while the exponent, also often denoted as the power, indicates how many times the base is multiplied by itself.

For example, \(5^3\) means that the base 5 is multiplied three times: \(5 \cdot 5 \cdot 5\) which equals 125. Exponentiation adheres to specific rules, such as the product of powers rule, which states that when multiplying two terms with the same base, you can add their exponents (\(a^m \cdot a^n = a^{m+n}\)).

However, with different bases such as in \(a^{3} \cdot b^{4}\), this rule cannot be applied. Also, when powers have different bases and exponents, as in our exercise, combining them through multiplication doesn't allow simplification. Understanding how exponents and powers work is essential for mastering algebraic manipulation and the simplification of expressions with powers.

Algebraic Rules for Simplification

Algebraic simplification is an orderly process of making an algebraic expression more accessible and manageable. The typical steps include combining like terms, using the distributive property, and applying the laws of exponents. Knowing basic algebraic rules for these operations is vital for simplification.

Some of these rules include the associative and commutative properties, which allow you to rearrange and group terms for easier combination. The distributive property makes it possible to multiply a term across a set of parentheses, as in \(a(b + c) = ab + ac\). Specific to exponents, there are rules for multiplying and dividing powers, taking powers of powers, and dealing with zero or negative exponents.

In the given exercise, \(a^{3} \cdot b^{4}\) does not allow for any further simplification based on these rules. The expression is already in its simplest form as the bases and exponents are different, providing no opportunity to use the algebraic rules that combine or reduce similar terms. This illustrates the importance of recognizing when an expression cannot be simplified just as much as knowing how to simplify it.