Problem 59 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ (4 x)^{-3} $$

Step-by-Step Solution

Answer

The expression \((4x)^{-3}\) with positive exponents is \(\frac{1}{64x^3}\)

Step 1: Apply negative exponent property

The first step is to apply the negative exponent property \(a^{-n} = \frac{1}{a^{n}}\). Using this, we can rewrite the expression \( (4x)^{-3} \) as \(\frac{1}{(4x)^{3}}\)

Step 2: Apply power of a product property

The next step is to apply the power of a product property. This property states that \((ab)^n = a^n b^n\), where \(a\) and \(b\) are the bases and \(n\) is the exponent. Applying this, we rewrite \(\frac{1}{(4x)^{3}}\) as \(\frac{1}{4^3 x^3}\).

Step 3: Calculate the power

To finish, calculate the power of the constant. In this case, the constant is 4, and its power is 3. Using exponent rules, we find that \(4^3 = 64\). So the final expression after simplification is \(\frac{1}{64x^3}\)

Key Concepts

Exponent Rules

Understanding exponent rules is essential for simplifying expressions with powers. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, when you see something like \(4^3\), it translates to \(4\times 4\times 4\). There are several key rules that help in managing exponents:

• The Product Rule: \(a^m \times a^n = a^{m+n}\) tells us that when we multiply two powers with the same base, we can add their exponents.
• The Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\) explains that when we divide two powers with the same base, we subtract the exponents.
• The Power of a Power Rule: \((a^m)^n = a^{mn}\) shows that when you raise a power to another power, you multiply the exponents.
• The Zero Exponent Rule: \(a^0 = 1\), provided \(aeq 0\), signifies that any base (except zero) raised to the power of zero equals one.
• The Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\) states that a negative exponent represents the reciprocal of the base raised to the opposite positive power.

This last rule is particularly pertinent to our exercise, where applying the negative exponent rule transformed \((4x)^{-3}\) into its reciprocal form \(\frac{1}{(4x)^3}\).

Powers of a Product Property

When dealing with powers of a product, remember the straightforward yet powerful property: \((ab)^n = a^n b^n\), where \(a\) and \(b\) are any real numbers, and \(n\) is an integer. This states that when you have a product raised to an exponent, each component of the product is raised to that exponent independently.

As it relates to our example, \((4x)^3\) involves both a number and a variable. According to the powers of a product property, this expression simplifies to \(4^3 x^3\), effectively distributing the exponent to both 4 and \(x\). This rule is vital for simplifying expressions and is especially useful in algebraic manipulations when variables are involved.

Simplifying Expressions

When it comes to simplifying expressions, the objective is to rewrite an expression in the most basic or compact form without changing its value. This process often involves a series of steps, including applying exponent rules and combining like terms.

Looking at our resolved exercise, the original form \((4x)^{-3}\) was not the simplest version. By recognizing and applying the correct rules, we achieved a more straightforward expression, \(\frac{1}{64x^3}\). This final expression is considered simplified as it no longer contains a negative exponent and has been reduced to a form where all the exponents are positive, making it easier for further algebraic operations.

Algebraic Manipulation

Algebraic manipulation is a critical skill within mathematics, referring to the process of rearranging and simplifying equations or expressions. This skillset involves being fluent with various arithmetic operations and algebraic properties, such as the distributive property, combining like terms, and of course, our exponent rules.

The ability to deftly manipulate algebraic expressions allows us to solve equations efficiently. For example, in the exercise, we employed algebraic manipulation to resolve \((4x)^{-3}\) into its simplest positive exponent form. This showcases how a combination of steps, starting with applying the negative exponent rule, then using the powers of a product property, and finally calculating the powers, enables us to transform and simplify algebraic expressions. Overall, mastering algebraic manipulation is fundamental to progress in higher-level math, as well as various applications of mathematics in real-world scenarios.