Problem 5 - 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: 4a^3 = 4a * a * a

Step 1: Identify the exponent in the given expression

In the expression \(4a^3\), the exponent is the number 3.

Step 2: Apply the exponent rule

The exponent (3) tells us how many times \(a\) is multiplied by itself. So, we need to rewrite the expression by multiplying \(a\) three times and keep the constant factor 4 unchanged.

Step 3: Rewrite the expression without exponent

Now, we can rewrite the expression as: $$4\cdot a \cdot a \cdot a$$

Step 4: Simplify the expression

Our final expression without exponents is: $$4a^3 = 4a\cdot a \cdot a$$

Key Concepts

Exponent Rules

Understanding exponent rules is essential when working with algebraic expressions, especially when simplifying or manipulating them. Exponents indicate how many times a number or variable is multiplied by itself. For example, in the algebraic expression \(4a^3\), the exponent is 3, which means \(a\) is to be multiplied by itself three times. Simplifying exponents involves replacing them with the expanded multiplication.

Key exponent rules include:

• The Product Rule: \(x^m \cdot x^n = x^{m+n}\), which says when you multiply two exponents with the same base, you add the exponents.
• The Power Rule: \(x^{m^n} = x^{m \cdot n}\), which indicates that when an exponent is raised to another exponent, you multiply the exponents.
• The Zero Exponent Rule: \(x^0 = 1\) for any number \(x\) except zero.
• The Negative Exponent Rule: \(x^{-n} = 1/x^n\), implying that negative exponents represent the reciprocal of the base raised to the positive exponent.

When these rules are applied correctly, they greatly simplify the process of working with algebraic expressions involving exponents.

Algebraic Expressions

Algebraic expressions are combinations of letters (variables), numbers, and at least one arithmetic operation. In the context of the given exercise, \(4a^3\) is an algebraic expression with \(4\) as the numerical coefficient and \(a^3\) as the variable part with an exponent. Simplifying an algebraic expression often involves combining like terms, applying exponent rules, and executing operations such as addition, subtraction, multiplication, or division.

When simplifying expressions, it’s essential to maintain the balance of the equation by performing the same operations on both sides if you're solving equations. For standalone expressions like \(4a^3\), simplification primarily focuses on reducing the expression to its most basic form, which involves expanding the exponent as seen in the solution steps.

Multiplication of Variables

Multiplication of variables follows the same basic principles as multiplication of numbers. In the given example, \(4a^3\), simplifying it involves expanding the exponent to show the repeated multiplication of the variable \(a\): \(4 \cdot a \cdot a \cdot a\). It's important to note that when variables are multiplied together, and they have the same base, their exponents should be added based on the Product Rule of exponents mentioned earlier.

When different variables are multiplied together, they are simply written next to each other, like \(ab\), which implies \(a\) multiplied by \(b\). If the same variable is multiplied, its exponents are summed, such as \(a^m \cdot a^n = a^{m+n}\). This simplification of exponent expressions helps in solving algebra problems more effectively and is a foundational skill in algebra.