Problem 49 - Algebra 2

Question

Simplify. $$ y^{2} z\left(y^{2} z^{3}-y z^{2}+3\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( y^{4} z^{4} - y^{3} z^{3} + 3 y^{2} z \).

Step 1: Distribute the Terms

Expand the expression by distributing the first term, \( y^{2}z \), to each term inside the parentheses. Do this to multiply each term individually:\[ y^{2}z \times (y^{2}z^{3}) - y^{2}z \times (yz^{2}) + y^{2}z \times 3 \]

Step 2: Simplify Each Distributed Term

Multiply the terms for each part of the expression one by one:- First Term: \( y^{2} z \times y^{2} z^{3} = y^{2+2} z^{1+3} = y^{4} z^{4} \)- Second Term: \( y^{2} z \times y z^{2} = y^{2+1} z^{1+2} = y^{3} z^{3} \)- Third Term: \( y^{2} z \times 3 = 3 y^{2} z \)

Step 3: Write the Simplified Expression

Now that each term has been simplified, write down the full expression:\[ y^{4} z^{4} - y^{3} z^{3} + 3 y^{2} z \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions. It involves spreading (or distributing) each term outside the parenthesis to every term inside. For example, when you have the expression \( y^2 z (y^2 z^3 - y z^2 + 3) \), you use the distributive property to multiply \( y^2 z \) with each term within the parentheses.

• Step-by-step: Multiply \( y^2 z \) with \( y^2 z^3 \), then with \( -y z^2 \), and finally with \( 3 \).
• This results in three separate multiplications: \( y^2 z \times y^2 z^3 \), \( y^2 z \times (-y z^2) \), and \( y^2 z \times 3 \).

By applying the distributive property, you transform a seemingly complicated expression into simpler pieces, which are easier to manage and solve step-by-step.

Exponent Rules

Exponent rules are essential when dealing with expressions that involve powers, as they guide us in multiplying and simplifying terms with exponents. These rules tell us how to handle multiplication between powers with the same base.

• Power Rule: When you multiply terms with the same base, you add their exponents. For instance, multiplying \( y^2 \) and \( y \) results in \( y^{2+1} = y^3 \).
• Example from Exercise: For the term \( y^2 z \times y^2 z^3 \), you apply the power rule by adding the exponents of alike bases: \( y^{2+2} = y^4 \) and \( z^{1+3} = z^4 \).

These rules simplify the process of dealing with large exponents by transforming multiplication into addition, making calculations friendlier and less error-prone. Once you're aware of exponent rules, expressions become considerably more approachable.

Multiplying Monomials

Multiplying monomials involves multiplying their coefficients (numbers) and applying exponent rules to variables.

• Understanding Monomials: A monomial is a single term consisting of numbers, variables, and exponents. For example, \( y^2 z \) is a monomial.
• Coefficient Multiplication: If monomials include coefficients, multiply them separately. In our exercise, \( y^2 z \times 3 \) involves multiplying the coefficient \(3\) with \( y^2 z \).
• Variable Handling: For \( y^2 z \times y^2 z^3 \), you handle the variables by multiplying like bases and adding exponents, resulting in \( y^4 z^4 \).

By mastering monomial multiplication, you can efficiently simplify expressions and tackle broader algebraic problems with confidence.