Problem 27 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{24z^4}\).

Step 1: Simplify individual terms

First, simplify the expression \((3z)^2\) which becomes \(3^2 imes z^2 = 9z^2\).Next, simplify \((6z^2)^{-3}\). The exponent \(-3\) means take the reciprocal and cube the terms:\((6z^2)^{-3} = \left(\frac{1}{6z^2}\right)^3 = \frac{1}{6^3z^6} = \frac{1}{216z^6}\).

Step 2: Multiply the simplified terms

To multiply the expressions \(9z^2\) and\(\frac{1}{216z^6}\), use the property of exponents \(z^a \cdot z^b = z^{a+b}\):\(9z^2 \times \frac{1}{216z^6} = \frac{9z^2}{216z^6}\).Simplify the fraction: \(\frac{9}{216} = \frac{1}{24}\).For the exponents, subtract: \(z^2 / z^6 = z^{2-6} = z^{-4}\).

Step 3: Write the final expression

The simplified expression is \(\frac{1}{24z^4}\). This result is achieved by combining our fraction result with the simplified exponent: \(\frac{1}{24}\) multiplied by \(z^{-4} = \frac{1}{z^4}\).Thus, the entire expression becomes \(\frac{1}{24z^4}\).

Key Concepts

Exponents

Understanding exponents is essential in simplifying expressions. An exponent refers to the number of times a number (the base) is multiplied by itself. For example,

• \(3^2\) means \(3 \times 3\), which equals 9.
• \(z^2\) implies \(z \times z\).

When dealing with expressions like \((3z)^{2}\), you apply the exponent to both the base number and the variable inside the parentheses. So \((3z)^{2}\) becomes \(3^2 \times z^2\), resulting in \(9z^2\).

Another concept to consider is the negative exponent. A negative exponent indicates dividing by that factor instead of multiplying. For example, \((6z^2)^{-3}\) requires using the reciprocal (as explained later) and taking the exponent of 3 for the denominator components.

Reciprocal

The reciprocal of a number is simply one divided by that number. It’s like flipping the number over. For example, the reciprocal of \(5\) is \(\frac{1}{5}\).

• If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).
• Realize that the reciprocal of a whole number \(x\) is \(\frac{1}{x}\).

In the expression \((6z^2)^{-3}\), the negative exponent means you need to take the reciprocal of \(6z^2\). This becomes \(\left(\frac{1}{6z^2}\right)^3\), turning the negative exponent into a positive one by making the base a fraction.
Understanding reciprocals is crucial when working with negative exponents as it helps in rewriting the expression in a form that is easier to simplify.

Multiplying Fractions

To multiply fractions, you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example,

• \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).

In the solution, after turning \((6z^2)^{-3}\) into \(\frac{1}{216z^6}\), we have to multiply it by \(9z^2\), written as \(\frac{9z^2}{1}\), yielding:

• \(\frac{9z^2}{1} \times \frac{1}{216z^6} = \frac{9z^2}{216z^6}\).

After multiplication, simplify the fraction.

Start with the coefficients: \(\frac{9}{216}\) simplifies to \(\frac{1}{24}\).Then subtract the exponents of \(z\): \(z^{2-6} = z^{-4}\), leaving the final expression in the simplified form of \(\frac{1}{24z^4}\). Working through each step with care leads to a clear and concise result.