Problem 30 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ 7^{3} \cdot 7^{0} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(7^{3} \cdot 7^{0}\) Answer: 343

Step 1: Identify the Product Rule of Exponents

The product rule for exponents states that when multiplying two numbers with the same base, simply add their exponents. It is given by: $$ a^{m} \cdot a^{n} = a^{m + n} $$ In our given expression \(7^{3} \cdot 7^{0}\), we have a base 7 raised to the power of 3 and 0. We will apply the product rule.

Step 2: Apply the Product Rule

Applying the product rule, we add the exponents like this: $$ 7^{3} \cdot 7^{0} = 7^{3 + 0} $$

Step 3: Simplify the Exponent

Performing the addition in the exponent, we get: $$ 7^{3 + 0} = 7^{3} $$

Step 4: Calculate the Result

Now we can calculate the value of the expression: $$ 7^{3} = 7 \cdot 7 \cdot 7 = 343 $$ So, the simplified expression is: \(343\).

Key Concepts

Product Rule for Exponents

When working with exponents, the product rule is a valuable tool for simplifying expressions. This rule applies when you are multiplying two expressions with the same base. Here, the base is the number that is raised to the power of an exponent. The product rule for exponents states that if you have two expressions with the same base, you can keep the same base and add the exponents together.

The product rule formula is:

• \(a^m \cdot a^n = a^{m+n}\)

This means if you are multiplying \(a^m\) by \(a^n\), you simply add \(m\) and \(n\).

In practice, this rule allows you to turn what could be a complex multiplication problem into a simple addition problem, at least as far as the exponents are concerned.

Simplification of Exponential Expressions

Simplification is a key skill in algebra and significantly helps in managing complex problems. When dealing with exponents, simplification often involves applying exponent rules like the product rule.

Consider an expression like \(7^3 \cdot 7^0\). Initially, this may appear challenging, but by applying the product rule we can quickly simplify it.

• Using the rule, combine the exponents: \(3 + 0 = 3\).
• Your expression \(7^3 \cdot 7^0\) simplifies to \(7^{3+0} = 7^3\).

Now, you're left with a much simpler exponential expression to evaluate. The process takes larger formulas and condenses them into more manageable forms, which is especially useful in further calculations or when solving equations.

Understanding Base and Exponent

In mathematics, the concept of exponents can drastically simplify the process of multiplication. Understanding the terms 'base' and 'exponent' is essential.

The base refers to the number that is being multiplied by itself. The exponent indicates how many times the base is used in the multiplication.

• In the expression \(7^3\), the number 7 is the base.
• The number 3 is the exponent, meaning you multiply the base, 7, by itself a total of three times: \(7 \cdot 7 \cdot 7\).

When the exponent is 0, like in \(7^0\), it follows a special rule: any number (except 0) raised to the power of 0 equals 1. This principle is helpful in simplifying many expressions and should always be remembered as a foundational aspect of exponent rules.