Problem 36 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(\left(\frac{5}{z}\right)^{2}\left(\frac{2}{z}\right)^{3}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(\frac{5}{z}\right)^{2}\left(\frac{2}{z}\right)^{3}\) is \(\frac{200}{z^{5}}\).

Step 1: Rewrite the Expression

Begin by writing the expression \(\left(\frac{5}{z}\right)^{2}\left(\frac{2}{z}\right)^{3}\) as \( \frac{5^{2}}{z^{2}} * \frac{2^{3}}{z^{3}} \), leveraging the law of exponents which states that \((a/b)^n = a^n / b^n\), where \(a\) is the base and \(n\) is the exponent.

Step 2: Compute the Numerators and Denominators

Next, calculate the numerators and denominators for each of the fractions individually. This leads us to \( \frac{25}{z^{2}} * \frac{8}{z^{3}} \), given that \(5^2 = 25\) and \(2^3 = 8\).

Step 3: Multiply the Fractions

Use the law of multiplication for fractions to multiply the fractions. We get \( \frac{25*8}{z^{2}*z^{3}} \).

Step 4: Simplify the Expression

Simplify the expression by performing the multiplication in the numerator and applying the exponent rules for the denominator to yield the simplified expression. The rule \(a^{m}*a^{n} = a^{m+n}\) is applied in the denominator. This results in \(\frac{200}{z^{5}}\).

Key Concepts

Law of Exponents

The law of exponents is crucial when working with algebraic expressions. It provides a set of rules that simplifies the process of multiplication, division, and raising powers to powers with the same base. For example, the rule \(a^m \cdot a^n = a^{m+n}\) tells us that when you multiply two exponents with the same base, you keep the base and add the exponents. This is the rule used to combine the denominators in the given problem \(\frac{5}{z}\right)^{2}\cdot\left(\frac{2}{z}\right)^3\), where both bases are \(z\). The exponents are added together (\(2+3\)), resulting in \(z^5\).
Understanding these rules makes computations much more manageable and helps to simplify expressions quickly and accurately.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it follows a simple and straightforward rule: Multiply the numerators (top numbers) to get the new numerator and multiply the denominators (bottom numbers) to get the new denominator. For instance, when given \(\frac{5^2}{z^2} \cdot \frac{2^3}{z^3}\), we multiply the numerators \(5^2\) and \(2^3\) to get \(25 \cdot 8\), and multiply denominators \(z^2\) and \(z^3\) to obtain \(z^{2+3}\) or \(z^5\). In the end, the fractions are combined to form a single fraction \(\frac{200}{z^5}\).
Remember that reducing fractions before multiplying can further simplify the process, although it wasn't necessary in this case.

Simplifying Numerators and Denominators

To simplify numerators and denominators effectively, first consider if there are any common factors that can be cancelled. In this problem, however, the focus is on handling exponents properly. For the given numerators, \(5^2\) simplifies to \(25\), and \(2^3\) simplifies to \(8\). The denominators, being the same variable raised to exponents, are combined under the law of exponents mentioned above.
Simplifying complex numerators and denominators often involves factoring or using other algebraic rules, but in the context of exponentiation, it's mainly about applying the right power rules to condense the expression.

Algebraic Rules and Properties

Algebra is built on a foundation of various rules and properties that govern how expressions are manipulated and simplified. These include distributive properties, commutative and associative properties, and operation rules for exponents and radicals. It's essential for students to become familiar with these properties to understand how to manipulate algebraic expressions correctly.
In our textbook example, we used these properties to rewrite the expression into a more manageable form before simplifying. Truly grasping these concepts not only helps solve textbook problems but also paves the way for success in more advanced mathematics.