Problem 43 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ (4 y)^{-3} $$

Step-by-Step Solution

Answer

The expression \((4y)^{-3}\) simplifies to \(\frac{1}{64y^3}\) with positive exponents.

Step 1: Understanding Negative Exponents

The expression given is \((4y)^{-3}\). In mathematics, a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. The formula to convert a negative exponent to a positive exponent is given by \(a^{-n} = \frac{1}{a^{n}}\).

Step 2: Apply the Negative Exponent Rule

Apply the negative exponent rule to the expression \((4y)^{-3}\). This means we take the reciprocal of \(4y\) and raise it to the power of 3: \((4y)^{-3} = \frac{1}{(4y)^3}\).

Step 3: Expand the Expression with Positive Exponents

Now, calculate \((4y)^3\): - First, raise 4 to the power of 3: \(4^3 = 4 \times 4 \times 4 = 64\)- Next, raise \(y\) to the power of 3: \(y^3\)Combine these results to get: \((4y)^3 = 4^3 \cdot y^3 = 64y^3\).

Step 4: Write the Expression in Simplest Form

Substitute the expanded form into the expression with positive exponents: \((4y)^{-3} = \frac{1}{64y^3}\).The expression is now in its simplest form with positive exponents only.

Key Concepts

Reciprocal

In the world of mathematics, understanding the concept of a reciprocal is essential, especially when dealing with negative exponents. The term "reciprocal" refers to the inverse of a number or expression.
For a given number or expression, say \( a \), its reciprocal is \( \frac{1}{a} \). This idea is crucial when simplifying expressions with negative exponents.

Here's how it works when encountering negative exponents:

• A negative exponent indicates that you need to find the reciprocal of the base.
• For example, if you have \( (4y)^{-3} \), the negative exponent tells us to take the reciprocal of \( (4y) \), leading to \( \frac{1}{(4y)^3} \).

By recognizing this and applying the reciprocal concept, you can effectively simplify and rewrite expressions to only include positive exponents. This approach not only makes expressions easier to handle but also aligns with the standard mathematics practices.

Simplifying Expressions

Simplifying expressions is a key skill in mathematics that helps to make calculations more manageable, and it's especially useful when working with exponents. The goal is to express the equation in its simplest and most straightforward form.
When simplifying, it's crucial to tackle one step at a time.

To simplify expressions involving negative exponents:

• Begin by applying the reciprocal to convert negative exponents into positive ones. For example, \( (4y)^{-3} \) becomes \( \frac{1}{(4y)^3} \).
• Next, calculate any powers. In the expression \( (4y)^3 \), start by computing \( 4^3\) and then \( y^3 \).
• Combine these results: \( 4^3 = 64 \) and \( y^3 \), to form \( \frac{1}{64y^3} \).

Through these steps, you convert the initial expression with negative exponents into its simplified equivalent with positive exponents.

Exponential Rules

Understanding exponential rules is critical for simplifying mathematical expressions. These rules dictate how to handle exponents, ensuring consistency and accuracy in calculations.
Here are some foundational rules to consider:

**Negative Exponent Rule**
This rule tells us that any base with a negative exponent can be rewritten with a positive exponent by taking the reciprocal of the base. Symbolically, if \( a^{-n} = \frac{1}{a^n} \), then \( (4y)^{-3} = \frac{1}{(4y)^3} \).

**Power of a Product Rule**
When an expression consists of several factors all raised to an exponent, apply the exponent to each factor separately:

• \((ab)^n = a^n b^n\)
• In our example, \((4y)^3 = 4^3 \cdot y^3 = 64y^3\)

**Combining Results for Simplification**
Utilize these rules to arrive at an expression that uses positive exponents only without additional operations or steps required. This results in cleaner, more interpretable expressions like \( \frac{1}{64y^3} \).
By mastering these rules, handling even the most complex exponential expressions becomes straightforward.