Problem 52 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ \left(a b c^{2}\right)^{3}\left(a^{2} b\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(a^{7}b^{5}c^{6}\)

Step 1: Apply Power to Power Rule

We can begin by applying the power-to-power rule, which states that \( (a^m)^n = a^{m*n} \). This rule needs to be applied to both \( (abc^2)^3 \) and \( (a^2b)^2 \). So we have \( a^{3}*b^{3}*c^{6} \) and \( a^{4}*b^{2} \).

Step 2: Apply Product of Powers Rule

Next, apply the product of powers rule which states that when multiplying with the same base, add the exponents. Applying the rule to \( a^{3}*a^{4} \) and \( b^{3}*b^{2} \), we get it simplified to \( a^{7} \) and \( b^{5} \). The term \( c^{6} \) remains as it is because there is no other term with the base 'c' to combine with.

Step 3: Combine the Powers

Combine all three terms to get the final answer. The simplified expression becomes \(a^{7}\)*\(b^{5}\)*\(c^{6}\).

Key Concepts

Power to Power Rule

The Power to Power Rule is a fundamental exponent rule in mathematics that simplifies expressions where powers are raised to other powers. When you have an expression like \((a^m)^n\), you can apply the Power to Power Rule to rewrite it as \(a^{m \times n}\). This means you multiply the exponents, rather than raising one power to another.
Imagine dealing with a term like \((abc^2)^3\). By applying the Power to Power Rule, each factor inside the parentheses is raised to the third power individually, so you get \(a^3\), \(b^3\), and \(c^{2 \times 3} = c^6\).
Similarly, for \((a^2b)^2\), applying the Power to Power Rule results in \(a^{2 \times 2} = a^4\) and \(b^2\).

• This rule is crucial for simplifying complex expressions efficiently without expanding them.
• Always remember to multiply the exponents when using this rule.

Understanding this rule helps unlock the door to solving various exponent-based problems with ease and precision.

Product of Powers Rule

The Product of Powers Rule is another essential tool for simplifying expressions involving exponents. This rule states that when you multiply two powers with the same base, you simply add the exponents. Specifically, \(a^m \times a^n\) equals \(a^{m + n}\).
Using this rule can greatly simplify your calculations, especially when handling expressions with several terms. For example, in the original problem, after using the Power to Power Rule, we had terms like \(a^3 \times a^4\) and \(b^3 \times b^2\).
Applying the Product of Powers Rule here, you sum the exponents for the base \(a\) resulting in \(a^{3+4} = a^7\). Similarly, for the base \(b\), you get \(b^{3+2} = b^5\).

• It’s important to note this rule only applies to terms sharing the same base.
• When terms do not share their base, you must treat each separately.

This rule is perfect for streamlining expressions and reducing complexity in solving exponent problems.

Combining Exponents

Combining exponents is the process of putting together the simplified parts of an expression that involve exponents to form a single, simplified expression. After using the Power to Power Rule and Product of Powers Rule, the next logical step is to combine these results to state the expression in its simplest form.
From our application of the Power to Power and Product of Powers rules, we are left with several simplified terms: \(a^7\), \(b^5\), and \(c^6\).
The final expression is therefore written as a single product of powers: \(a^7 \times b^5 \times c^6\).
This process is vital in mathematics as it allows you to present complex expressions in a reduced and more understandable form.

• Combining exponents logically follows simplifying each base with the previous rules.
• This step involves no additional mathematical computation but requires accuracy in arranging the terms.

Thus, with all powers combined seamlessly, you arrive at a tidy and elegant mathematical expression ready for interpretation or further application.