Problem 43 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ (3 z)^{2}\left(6 z^{2}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{24z^4}\).

Step 1: Expand the Expression

First, expand the given expression: \[(3z)^2(6z^2)^{-3}.\]This involves raising each part of the terms inside the parentheses to the given power.

Step 2: Simplify Each Part

Calculate \((3z)^2 = 3^2 \cdot z^2 = 9z^2\)and \((6z^2)^{-3} = 6^{-3} \cdot (z^2)^{-3} = \frac{1}{6^3} \cdot \frac{1}{z^6} = \frac{1}{216z^6}\).

Step 3: Multiply the Results

Multiply the terms obtained from Step 2: \(9z^2 \times \frac{1}{216z^6}\).

Step 4: Simplify Fractions and Powers of z

Simplify the fraction and the powers of \(z\): \[\frac{9z^2}{216z^6} = \frac{9}{216} \cdot \frac{z^2}{z^6} = \frac{1}{24} \cdot z^{-4}.\] Thus, we have \(\frac{1}{24z^4}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing complex math problems into simpler forms. This process makes it easier to work with equations and perform calculations. In the context of exponentiation, each component of the expression must be handled separately before combining them back together. The steps include

• expanding expressions by applying the exponents to both coefficients and variable parts,
• simplifying numerical constants,
• and reducing variable parts by managing their associated powers.

Generally, you want to follow the order of operations and methodically manage each part of the expression. By simplifying, you turn a complicated expression into something more manageable, making further manipulation much more straightforward.

Negative Exponents

Negative exponents can seem tricky at first, but they simplify to something familiar when you understand the concept. A negative exponent indicates how many times you need to divide by the base, rather than multiply. For example,

• if you see a term like \(a^{-b}\), it transforms into \(\frac{1}{a^b}\),
• converting division back into multiplication in the fraction form.

This is why \((6z^2)^{-3}\) becomes \(\frac{1}{6^3} \cdot \frac{1}{z^6}\). When simplifying an expression, treat the negative exponent by flipping the base to its reciprocal position, turning potential negatives into manageable fractions. By understanding this property, working through exponentiation problems with negative exponents becomes significantly more intuitive.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent a particular mathematical problem or scenario. In our exercise, the expression is \((3z)^2(6z^2)^{-3}\). Here, we deal with coefficients (3 and 6) and the variable (z) raised to various powers. The key aspects of manipulating algebraic expressions include:

• applying rules of exponents efficiently where you multiply powers,
• divide powers when necessary,
• and reorder terms to achieve the simplest form.

In this example, cubing a constant means multiplying by itself three times (as seen with \(6^{-3}\)), and raising a variable to a negative power means flipping it to the denominator. Eventually, through controlled steps, the expression simplifies to \(\frac{1}{24z^4}\), combining all elements cohesively for simplicity and clarity.