Problem 56 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(\left(6 z^{4}\right)^{2} \cdot z^{3}\)

Step-by-Step Solution

Answer

The simplified form of the expression given is \( 36z^{11} \).

Step 1: Apply the power-of-a-power rule

By using the power-of-a-power rule, which says that \( (a^{m})^{n} = a^{m*n} \), we can rewrite \( (6z^{4})^{2} \) as \( 6^{2} \cdot (z^{4})^{2} = 36z^{8} \).

Step 2: Multiply the simplified expression with \( z^{3} \)

Now, simply multiply \( 36z^{8} \) by \( z^{3} \) to continue the simplification process leading us to \( 36z^{8} \cdot z^{3} \).

Step 3: Apply the exponent rule for multiplication

Now, we use the exponent rule that states \( a^{n}\cdot a^{m} = a^{n+m} \) to simplify \( z^{8} \cdot z^{3} \) to \( z^{11} \). Thus, the entire equation simplifies to \( 36z^{11} \).

Key Concepts

Understanding the Power-of-a-Power Rule

When simplifying algebraic expressions, one common hurdle is managing exponents, particularly when they're nested within each other. This is where the power-of-a-power rule becomes invaluable. It states that when you have a power raised to another power, like \( (a^m)^n \), you can simplify it by multiplying the exponents, resulting in \( a^{m*n} \).

Let's consider the expression \( (6z^4)^2 \). Applying the power-of-a-power rule, the base, 6, is raised to the 2nd power, becoming 36. The variable \( z \) is also affected: its exponent, 4, is multiplied by 2, resulting in an exponent of 8. Consequently, this rule simplifies the expression to \( 36z^8 \), making it much easier to work with in subsequent steps.

Applying the Exponent Multiplication Rule

Further simplification of expressions often involves combining like terms, especially when they are part of multiplication. This is where the exponent multiplication rule comes to play. This rule is simple: when you multiply two expressions with the same base but different exponents, you add the exponents. In mathematical terms, \( a^n \cdot a^m = a^{n+m} \).

In our example, after applying the power-of-a-power rule, we were left with \( 36z^8 \). Multiplying that by \( z^3 \) requires us to add the exponents of \( z \) due to the exponent multiplication rule. We have \( z^8 \) and \( z^3 \), so we add the 8 and 3 together, giving us \( z^{8+3} = z^{11} \). This simplification allows us to combine these terms into a single expression, \( 36z^{11} \), providing a neat and comprehensible result.

Mastering Algebraic Manipulation

Algebraic manipulation is an umbrella term that includes various techniques to simplify or rearrange algebraic expressions. From combining like terms to factoring, it encapsulates all methods aimed at making expressions easier to understand or solve.

In the given exercise, we used a combination of algebraic manipulation techniques, including the power-of-a-power rule and the exponent multiplication rule. Key to mastering algebraic manipulation is understanding how different rules interact. For instance, recognizing when to apply the power-of-a-power rule before combining terms with the exponent multiplication rule is crucial for correct simplification.

By practicing these algebraic manipulation techniques often and in different combinations, you will develop a stronger intuition for simplification and problem-solving in algebra.