Problem 25 - Algebra and Trigonometry

Question

Simplify each expression. $$ (-3 y)^{4} $$

Step-by-Step Solution

Answer

The expression \((-3y)^4\) simplifies to \(81y^4\).

Step 1: Understand Exponentiation

When you have an expression like \((-3y)^4\), the exponent 4 means you are multiplying \(-3y\) by itself four times. So, \((-3y)^4\) can be expanded to \((-3y)\times(-3y)\times(-3y)\times(-3y)\).

Step 2: Apply Exponent Rules

According to exponent rules, \((ab)^n = a^n b^n\). Here, \(a = -3\) and \(b = y\), so we apply the exponent to both components: \((-3)^4\times y^4\).

Step 3: Calculate \((-3)^4\)

Calculate \((-3)^4\) by multiplying \(-3\) four times: \(-3 \times -3 = 9\), \(9 \times -3 = -27\), and \(-27 \times -3 = 81\). So, \((-3)^4 = 81\).

Step 4: Combine Results

Now, combine the results from Step 2 and Step 3: \(81\times y^4\).

Step 5: Final Simplified Expression

The expression \((-3y)^4\) simplifies to \(81y^4\).

Key Concepts

Understanding Exponentiation

Exponentiation is the process of raising a number or an expression to a power. This power, also known as the exponent, tells you how many times to multiply the base by itself. For example, in the expression \((-3y)^4\), the base is \((-3y)\) and the exponent is \(4\). This means you need to multiply \(-3y\) by itself four times, resulting \[(-3y) \times (-3y) \times (-3y) \times (-3y).\]When simplifying expressions \(((x^n)\)), focus on correctly expanding the multiplication, as it forms the foundation for further calculations. This is particularly crucial when the base is a composite term like \(-3y\), since both the number \((-3)\) and the variable \(y\) must be considered in the process.

Applying Exponent Rules

Exponent rules are essential tools that help simplify expressions. One important rule is \((ab)^n = a^n \times b^n\).This means that when raising a product to a power, each factor in the product should be individually raised to that power. In our example of \((-3y)^4\), this rule lets us separately calculate the powers for

• The number \((-3)\) raised to the fourth power, and
• The variable \(y\) also raised to the fourth power.

After applying this rule, the original expression \((-3y)^4\) becomes \((-3)^4 \times y^4\). This simplification makes calculating the expression more straightforward and organized.

Dealing with Negative Numbers

Understanding how negative numbers behave when used with exponents can minimize errors in calculations. A negative base raised to an even exponent results in a positive number. This is because the negative signs effectively "cancel" each other when multiplied an even number of times.In our step-by-step approach, \((-3)^4 = 81\).The reason for this positivity is clear when breaking down each multiplication step:

• \(-3 \times -3 = 9\), both negative signs cancel to a positive.
• Then, \(9 \times -3 = -27\), introducing another negative sign.
• Finally, \(-27 \times -3 = 81\), with the final negative turned positive.

Getting comfortable with negative base exponentiation can clear up confusion when simplifying similar expressions. Always check how the sign changes with each multiplication step.