Problem 50 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(a b^{3} c^{2}\right)^{5}\left(a^{2} b^{2} c\right)^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(a b^{3} c^{2}\right)^{5}\left(a^{2} b^{2} c\right)^{2}\). Answer: \(a^9b^{19}c^{12}\)

Step 1: Applying the power rule to each term

First, let's apply the power exponent rule: \({(x^a)^b} = x^{a \cdot b}\) to each term in the expression. This means we need to multiply the exponent of each term by the exponent of the brackets. $$ \left(a b^{3} c^{2}\right)^{5}\left(a^{2} b^{2} c\right)^{2} = a^{1\cdot5}b^{3\cdot5}c^{2\cdot5}a^{2\cdot2}b^{2\cdot2}c^{1\cdot2} $$

Step 2: Simplifying the obtained expression

Now, simplify the expression by multiplying the exponents: $$ = a^{5}b^{15}c^{10}a^{4}b^{4}c^{2} $$

Step 3: Combining like terms

Combine like terms by adding their exponents: $$ = a^{(5+4)}b^{(15+4)}c^{(10+2)} = a^9b^{19}c^{12} $$ The simplified expression is \(a^9b^{19}c^{12}\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is akin to tidying up a room: you organize and combine similar items to make it easier to understand what you have. In algebra, this process involves combining like terms, which are terms that have the exact same variables raised to the same power. Like terms can be added or subtracted from each other.

To simplify an algebraic expression, follow these steps:

• Identify like terms.
• Rearrange the expression to place like terms together, which is not mandatory but helps with clarity.
• Combine the coefficients of like terms.
• Apply the rules of exponents when dealing with variables raised to a power.

Considering the original exercise \(\left(a b^{3} c^{2}\right)^{5}\left(a^{2} b^{2} c\right)^{2}\), the expression was tidied by first expanding the powers using exponent rules and then combining like terms to arrive at the final simplified form \(a^9b^{19}c^{12}\).

Applying Exponent Rules

Exponent rules, also known as laws of exponents, are essential tools for simplifying expressions involving powers. One such rule is the power of a power rule, represented as \( (x^a)^b = x^{a \cdot b} \). This rule states that when you raise a power to another power, you multiply the exponents.

Let's apply this to our exercise. We took the expression \(\left(a b^{3} c^{2}\right)^{5}\left(a^{2} b^{2} c\right)^{2}\) and used the power of a power rule to expand each term. For example, the term \(b^{3}\) raised to the 5th power became \(b^{3\cdot5}\), which simplified to \(b^{15}\). It's important to apply these rules separately for each base variable within an expression to ensure it simplifies correctly.

Combining Like Terms

Once exponent rules have been applied, and we have an expanded expression involving powers, the next step is to combine like terms to further simplify the expression. Like terms are terms that have exactly the same variable part. You can combine like terms by adding their exponents.

For instance, in the given example we had the terms \(a^{5}\) and \(a^{4}\). Since they have the same base \(a\), we can add the exponents to get \(a^{5+4}\) which simplifies to \(a^9\). This process was repeated for the other variables in the expression. When combining like terms, remember not to mix different bases, since only terms with the same base can be combined in this manner.