Problem 6 - Elementary Algebra

Question

Write each of the following without exponents. $$ (4 a)^{3} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \((4a)^3\) without using exponents. Answer: The expression \((4a)^3\) rewritten without exponents is \(64a^3\).

Step 1: Apply the exponent rule

Using the exponent rule \((ab)^c = a^c \cdot b^c\), we can rewrite the given expression as: $$ (4a)^3 = 4^3 \cdot a^3 $$

Step 2: Calculate the value of \(4^3\)

Compute the value of the numeric part, which in our case is \(4^3\), as follows: $$ 4^3 = 4 \cdot 4 \cdot 4 = 64 $$

Step 3: Rewrite the expression without exponents

Now replace \(4^3\) with its calculated value and rewrite the expression: $$ (4a)^3 = 64 \cdot a^3 $$

Step 4: Final Answer

Therefore, the expression \((4a)^3\) without exponents is \(64a^3\).

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables (letters), and operation symbols. It represents a value or a set of values for which the expression is defined. For example, in the expression \(4a^3\), \(4\) is a coefficient (a numerical factor), \(a\) is a variable representing an unknown value, and \(3\) is the exponent indicating how many times \(a\) is used as a factor.

An important aspect of algebraic expressions is that they can be manipulated using various algebraic rules to simplify or transform them into different forms, allowing for easier interpretation and solving of mathematical problems. Simplifying these expressions involves performing operations like addition, subtraction, multiplication, division, and applying exponent rules, which provides us with a more concise version of the original expression that is equivalent.

Simplifying Exponents

Simplifying exponents, also known as 'simplifying powers', is an essential skill when dealing with algebraic expressions. Exponent rules help us to simplify expressions with exponents in a systematic and logical way. Here are the fundamental rules for simplifying exponents:

• \textbf{Product Rule}: \(a^m \times a^n = a^{m+n}\), where \(a\) is the base, and \(m\) and \(n\) are the exponents.
• \textbf{Quotient Rule}: \(a^m \text{ / }\ a^n = a^{m-n}\) when \(m > n\).
• \textbf{Power of a Power Rule}: \(\left(a^m\right)^n = a^{mn}\), which states that when you raise a power to another power, you multiply the exponents.
• \textbf{Power of a Product Rule}: \(\left(ab\right)^n = a^n \times b^n\), which means that when two or more factors are raised to a power, each factor is raised to the power separately.
• \textbf{Negative Exponent Rule}: \(a^{-n} = \frac{1}{a^n}\), which tells us that negative exponents represent the reciprocal of the base raised to the positive exponent.

Applying these rules can greatly simplify the process of working with algebraic expressions containing exponents, often making it possible to rewrite them without exponents entirely, or express them in a simpler form.

Exponential Notation

Exponential notation is a compact way to represent repeated multiplication of the same factor. It involves two key components: the base and the exponent. The base is the number that is being multiplied, and the exponent tells us how many times the base is used as a factor.

For instance, \(4^3\) is an exponential notation where \(4\) is the base, and \(3\) is the exponent. It signifies that \(4\) is multiplied by itself three times. Hence, \(4^3 = 4 \times 4 \times 4 = 64\).

Understanding exponential notation is crucial in algebra and higher mathematics, as it is extensively utilized to express large numbers, model exponential growth or decay, and represent powers and roots. As shown in the exercise, when simplifying expressions, knowing how to convert between exponential form and expanded form can make it easier to calculate and understand the value of the expression.