Problem 37 - Algebra 1

Question

Simplify the variable expression. $$-\left(y^{4}\right)(y)$$

Step-by-Step Solution

Answer

The simplified variable expression is \( -y^{5} \).

Step 1: Identify the base and the exponents

In this expression, the base is \( y \) and it has exponents \( 4 \) and \( 1 \) respectively.

Step 2: Apply the Rule of Multiplying Powers

This rule states that when multiplying the powers with the same base, their exponents will add up. Here, the exponents are \( 4 \) and \( 1 \). So, addition will result in \( 4 + 1 = 5 \).

Step 3: Write down the simplified expression

The simplified variable expression is \( -y^{5} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing a complex expression into its simplest form. This process makes calculations easier and the expression more comprehensible. In many cases, you do this by applying various algebraic rules and properties.

For the given example, we start with the expression \(-\left(y^{4}\right)(y)\). Our goal is to simplify it to something more manageable. Simplifying doesn’t alter the expression’s value, but makes it easier to work with in future calculations.

The process involves identifying parts of the expression that can be combined and using algebraic rules to perform these operations. In our case, we look at the exponents of the variable \(y\) and apply exponent rules to combine them efficiently. After simplifying, we arrive at \(-y^{5}\), which is much easier to interpret.

Multiplying Powers

Multiplying powers with the same base involves adding their exponents. This is a fundamental exponent rule that allows you to streamline expressions with ease.

Here’s how it works: Suppose you have an expression with two powers of the same base, like \(y^{4}\) and \(y^1\). When you multiply these, you add their exponents together.

• Base: The core value being raised to a power, e.g., \(y\).
• Exponents: The numbers that indicate how many times the base is multiplied by itself.

In this example, the base \(y\) has exponents \(4\) and \(1\). Using the rule of multiplying powers, we add these exponents together: \(4 + 1 = 5\). The expression becomes \(y^{5}\), which is the simplified form. This rule is particularly useful when dealing with variables and can significantly cut down on cumbersome calculations.

Variables with Exponents

Variables with exponents represent repeated multiplication of the variable by itself. Exponents can make dealing with larger expressions straightforward by condensing the expression into a more compact form.

In algebra, you’ll often encounter variables like \(y\) raised to various powers. Each exponent tells you how many times \(y\) is used as a factor. For example, \(y^4\) means \(y\times y\times y\times y\).

Understanding how exponents work with variables is crucial for simplification and performing algebraic operations efficiently. When variables have the same base, the exponent rules—such as multiplying powers by adding the exponents—allow you to combine them easily. This principle keeps expressions tidy and manageable. In our example, combining \(y^4\) and \(y^1\) under multiplication becomes a matter of adding their exponents, simplifying to \(y^5\). This clarity aids significantly in both basic and advanced algebraic manipulations.