Problem 39 - Precalculus Mathematics for Calculus

Question

Simplify each expression. (a) \(\left(a^{2} a^{4}\right)^{3}\) (b) \(\left(\frac{a^{2}}{4}\right)^{3}\) (c) \((3 z)^{2}\left(6 z^{2}\right)^{-3}\)

Step-by-Step Solution

Answer

(a) \(a^{18}\), (b) \(\frac{a^6}{64}\), (c) \(\frac{1}{24z^4}\)

Step 1: Apply Power Rule to Part (a)

For the expression \((a^{2} a^{4})^{3}\), first apply the power of a product rule, which states \((x^m x^n)^p = x^{m*p+n*p}\). This simplifies the expression to: \[ (a^{2+4})^3 = (a^6)^3 \] Now apply the power of a power rule: \((x^m)^n = x^{m*n}\), and simplify to \( a^{18} \).

Step 2: Simplify Part (b) Using Power Rule

For the expression \(\left(\frac{a^2}{4}\right)^3\), we apply the power to both the numerator and denominator separately: \(\frac{a^{2*3}}{4^3}\)This simplifies to:\[ \frac{a^6}{64} \]

Step 3: Simplify Part (c) Using Combined Index Laws

For \((3z)^2(6z^2)^{-3}\), apply the power rule \((x^m)^n = x^{m*n}\) to each term individually:1. \((3z)^2 = 3^2 \cdot z^2 = 9z^2\)2. \((6z^2)^{-3} = \frac{1}{(6^3 \cdot z^{2*3})} = \frac{1}{216z^6} \)Now combine:\[ 9z^2 \cdot \frac{1}{216z^6} = \frac{9}{216} \cdot z^{2-6} \]This simplifies to: \( \frac{1}{24}z^{-4} \) or \( \frac{1}{24z^4} \)

Key Concepts

Power Rule Explanation

The power rule is a convenient tool in algebra for handling expressions that involve exponents. This rule is especially useful when you need to simplify expressions raised to another power. The power rule states that when you have a term in the form

• \((x^m)^n\),
• it simplifies to \(x^{m \cdot n}\).

This essentially means you multiply the exponents. For example, with the expression \((a^6)^3\), applying the power rule gives you \(a^{18}\).

Another version of the power rule involves several multiplied bases, such as \((x^m x^n)^p\). According to this rule, you apply it as \((x^m x^n)^p = x^{m \cdot p + n \cdot p}\). This allows you to handle multiple terms at once while applying the power. Let’s say you have \((a^2 a^4)^3\); using the power rule, it becomes \(a^{18}\).

Understanding the power rule is essential for forming a strong foundation in algebra and makes working with exponents much simpler.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that makes working with equations easier and neater. Simplification involves reducing an expression to its simplest form while maintaining the same value.

When working with expressions involving exponents, like \((3z)^2(6z^2)^{-3}\), you often need to deal with them separately. Breaking it down, begin by simplifying each component:

• Apply the power rule to get \(9z^2\) from \((3z)^2\).
• For \((6z^2)^{-3}\), compute \(1/(216z^6)\).

After simplifying each separate expression, you then combine them:

• Multiply their coefficients and adjust the exponents by adding or subtracting where needed, e.g., \(9z^2 \cdot \frac{1}{216z^6} = \frac{1}{24z^4}\) or \(\frac{1}{24}z^{-4}\).

This process of simplifying expressions using consistent rules helps streamline problem solving in algebra.

Index Laws in Algebraic Expressions

Index laws, also known as laws of exponents, are rules that describe how to handle math operations involving powers. They are crucial for simplifying expressions efficiently.

• The basic laws include rules for multiplying and dividing powers with the same base, raising a power to another power, and handling negative and zero exponents.

Consider a scenario like \(\frac{a^6}{64}\). Here, you’re dealing with an expression modified by exponents both in the numerator and the denominator. Applying the index laws allows you to get \(\frac{a^6}{4^3}\), which simplifies to \(\frac{a^6}{64}\).

Moreover, with expressions like \(z^{2-6}\) that emerge from multiplying or dividing powers, index laws help you identify that the factor can simplify further if possible:

• Subtract the exponents, yielding \(z^{-4}\), and express as \(\frac{1}{z^4}\) if preferred, making it easier to interpret.

Understanding and applying index laws prevent errors and streamline calculations involving exponents in algebraic expressions.