Problem 57 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left(y^{5}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(y^{15}\).

Step 1: Identify the Power of a Power Rule

The power of a power rule states that when you raise a power to another power, you multiply the exponents. This rule is written as \((a^m)^n = a^{m\cdot n}\).

Step 2: Apply the Rule to the Given Expression

Apply the power of a power rule to the expression \((y^5)^3\). Here, the base \(y\) has an exponent of 5, and this whole expression is raised to the power of 3. Therefore, we calculate it as \(y^{5\cdot 3}\).

Step 3: Multiply the Exponents

Multiply the exponents 5 and 3: \[5 \cdot 3 = 15\] Hence, the expression simplifies to \(y^{15}\).

Step 4: Write the Final Simplified Expression

After multiplying the exponents, the simplified expression is \(y^{15}\).

Key Concepts

Exponents

Exponents are a mathematical notation indicating the number of times a number is multiplied by itself. It's like shorthand for repeated multiplication. For example, if you see the notation \(y^5\), it means \(y\) is multiplied by itself 5 times, or \(y \times y \times y \times y \times y\). Using this simple concept, mathematicians have developed various rules to simplify complex expressions involving exponents. An exponent has two components: the base (in this case, \(y\)) and the exponent itself (the number \(5\) in \(y^5\)). Together, they help express large numbers in compact form.Understanding the basics of exponents is crucial for tackling more advanced algebraic concepts, such as simplifying expressions and working with polynomial equations. So remember, whenever you see an exponent, think of it as a helpful tool for efficient calculation and representation.

Simplifying Expressions

Simplifying expressions involves using mathematical rules and operations to rewrite expressions in their simplest form. This process often includes reducing or combining terms to make calculations easier and results more concise. When working with exponents, simplifying can involve several approaches:

• Combining like terms
• Applying arithmetic operations
• Using specific rules for exponents, such as the power rule

The goal is to make expressions more manageable while retaining their original value. For example, when you apply the power of a power rule to simplify \((y^5)^3\), you multiply the exponents instead of expanding the expression as \(y \times y \times \, ... \, (15 \, times)\). This approach makes it far easier to handle and visualize.Not only does simplifying expressions make them more aesthetically pleasing, but it also aids in performing further algebraic operations and solves complex problems efficiently.

Power of a Power Rule

The power of a power rule in exponents is a fundamental concept that simplifies managing expressions with multiple layers of exponents. Imagine stacking powers; instead of dealing with a series of multiplications, this rule lets you cut straight to the chase.The power of a power rule states that when you raise a power to another power, you multiply the exponents. This can be represented as \((a^m)^n = a^{m \cdot n}\). Let's break this down:

• \(a\) is the base
• \(m\) is the first exponent
• \(n\) is the second exponent

In the exercise \((y^5)^3\), applying the power of a power rule, you multiply \(5\) by \(3\) to get \(15\). Thus, the expression simplifies to \(y^{15}\). This shortcut keeps the expression neat and manageable without expanding it fully.Mastering the power of a power rule allows you to handle more complex mathematical scenarios efficiently, transforming what might seem like a daunting task into a straightforward process.