Problem 48 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(8 a^{2} z\right)\left(\frac{1}{2} a^{3} z^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(4a^{5}z^{5}\).

Step 1: Multiply the Coefficients

First, multiply the coefficients of the terms. The coefficients here are 8 and \(\frac{1}{2}\). Calculate \(8 \times \frac{1}{2} = 4\).

Step 2: Add the Exponents of 'a'

Identify and add the exponents of \( a \). In the original expression, \( a^{2} \) and \( a^{3} \) appear. Use the property \( a^{m} \times a^{n} = a^{m+n} \) to add the exponents: \( 2 + 3 = 5 \). Thus, you have \( a^{5} \).

Step 3: Add the Exponents of 'z'

Identify and add the exponents of \( z \). Here you have \( z^{1} \) from the first term and \( z^{4} \) from the second term. Add the exponents: \( 1 + 4 = 5 \). Thus, you have \( z^{5} \).

Step 4: Construct the Simplified Expression

Combine the results from the previous steps to form the simplified expression. You have a coefficient of 4, \( a^{5} \), and \( z^{5} \). Thus, the expression simplifies to \( 4a^{5}z^{5} \).

Key Concepts

Multiplying Coefficients

When simplifying algebraic expressions, one of the first things you often need to do is multiply the coefficients. Coefficients are the numbers in front of the variables, and they tell us how many times the variable is being taken. In our exercise, the coefficients are 8 and \( \frac{1}{2} \). To multiply these, you simply perform regular multiplication:

• Multiply 8 by \( \frac{1}{2} \) to get 4.

This step is crucial because it simplifies the expression right from the start by reducing it to a more manageable form. Always begin by multiplying the coefficients when the terms are in a multiplication relationship.

Adding Exponents

Exponents are a way to express repeated multiplication of the same number or variable. When dealing with variables with exponents in multiplication, you should add the exponents. The rule is:

• \( a^{m} \times a^{n} = a^{m+n} \)

In our exercise, the variable 'a' appears as \( a^{2} \) and \( a^{3} \). Adding their exponents gives \( a^{2+3} = a^{5} \).

• For the 'z' variables: \( z^{1} \times z^{4} = z^{1+4} = z^{5} \)

Adding exponents is a powerful tool because it allows you to combine like terms quickly and efficiently, streamlining the algebraic expression.

Simplifying Expressions

Simplifying expressions means making them easier to understand or use by minimizing complex factors. After multiplying coefficients and adding exponents, you are ready to simplify the entire expression. Bring all the elements together:

• The coefficient we found, which is 4.
• The simplified powers of the variables, \( a^{5} \) and \( z^{5} \).

Combine these to get the final simplified expression: \( 4a^{5}z^{5} \). This simple, clean form makes it easier to see the relationships between components and to use the expression in further calculations or applications. Always aim to simplify expressions for clarity and ease of use in both practical and theoretical applications.