Problem 15 - Beginning and Intermediate Algebra

Question

Simplify. $$y^{3} \cdot y^{-7}$$

Step-by-Step Solution

Answer

The short answer is: \(y^{3} \cdot y^{-7} = y^{-4}\).

Step 1: Identify the rule for multiplying exponents with the same base

When multiplying two exponents with the same base, we can add the powers together. In this case, we have \(y^3\) times \(y^{-7}\), and we can use this rule to simplify the expression: \(y^{m} \cdot y^{n} = y^{m+n}\).

Step 2: Add the exponents together

Using the rule from Step 1, we can add the exponents of our expression together: \(y^{3} \cdot y^{-7} = y^{3 + (-7)}\).

Step 3: Simplify the expression

Now, simply add the exponents together to get the simplified expression: \(y^{3 + (-7)} = y^{-4}\). So, the simplified expression is \(y^{-4}\).

Key Concepts

Algebra

Algebra is more than just numbers and variables. It's a branch of mathematics that uses symbols and letters to represent numbers and express mathematical relationships. Variables like \( y \) in our exercise can take various values, allowing algebra to solve equations and express general principles.

In algebra, we perform operations such as addition, subtraction, multiplication, and division on these variables and expressions to simplify or solve them. The essence of algebra is to find unknown values and understand the relationships between different quantities. To do this, we rely on rules that guide us in manipulating these symbols, like the exponent rules used in our exercise.

Multiplying Exponents

When we multiply exponents with the same base, such as \( y^3 \) and \( y^{-7} \), the exponent rule tells us to simply add the exponents. This rule emerges because of the way exponents represent repeated multiplication.

Let's break it down:

• \( y^3 \) means \( y \times y \times y \)
• \( y^{-7} \) means \( \frac{1}{y^7} \)

When you multiply \( y^3 \) by \( y^{-7} \), we are adding the exponents: \( y^{3 + (-7)} = y^{-4} \). Adding exponents makes it easier to express the multiplication of powers with the same base compactly.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This often makes them easier to understand or solve. When you simplify, you adhere to established mathematical rules and properties.

In our example, \( y^{3} \cdot y^{-7} \) becomes \( y^{-4} \) through the process of combining like terms using exponent rules. By adding \( 3 \) and \( -7 \), we find the expression more manageable and concise: \( y^{-4} \).

This simplification process is crucial in algebra as it helps prevent errors, streamlines calculations, and leads to more straightforward problem-solving steps. Always aim for a form that follows the rules of mathematics while maintaining the original intent of the expression.