Problem 64 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(3^{4}\right)\left(7^{0}\right)(2) $$

Step-by-Step Solution

Answer

The expression simplifies to 162.

Step 1: Apply the Zero Exponent Rule

The Zero Exponent Rule states that any nonzero base raised to the power of zero is equal to one. Therefore, simplify the expression by converting \(7^0\) to 1. Your expression now becomes: \[ (3^4)(1)(2) \]

Step 2: Simplify the Expression

Now, we'll simplify the expression by first calculating \(3^4\). Calculate \(3^4\):\[ 3^4 = 3 \times 3 \times 3 \times 3 = 81 \]Now substitute back into the expression:\[ 81 \times 1 \times 2 = 81 \times 2 \]

Step 3: Calculate the Final Result

Calculate the product:\[ 81 \times 2 = 162 \].Thus, the simplified expression using positive exponents only is 162.

Key Concepts

Zero Exponent Rule

When working with exponents, one important rule to remember is the **Zero Exponent Rule**. This rule states that any non-zero number raised to the power of zero equals one. For example, if you see an expression like \(7^0\), you can simplify it immediately to 1. This is because any base, when raised to the power zero, essentially reduces to 1.
Understanding this rule can greatly simplify complex expressions by reducing elements. Imagine you're working with a long algebraic expression full of various exponents. If you encounter any term with a zero exponent, you can make your life easier by converting it to 1 right away.
This principle is crucial in algebra because it helps to quickly eliminate terms and simplify expressions without affecting the resultant value. Remember, though, the base should be non-zero for this rule to hold true.

Simplifying Expressions

In mathematics, simplifying an expression means reducing it to its simplest form. This involves combining like terms and breaking down expressions to make them easier to read and solve.
Let's take a look at how we simplified the expression \((3^4)(7^0)(2)\) from the original exercise. After applying the Zero Exponent Rule and converting \(7^0\) to 1, our expression became \((3^4)(1)(2)\).
Next, we calculated \(3^4\), which is the expansion of the multiplication \(3 \times 3 \times 3 \times 3 = 81\). Once we arrived at the value of 81, it was substituted back into the expression.

• The expression now reads as \(81 \times 1 \times 2\).
• Multiplying 81 by 1 doesn't change the value, so you're left with \(81 \times 2\).
• Finally, completing the multiplication gives you 162.

This results in a simplified expression that is easy to understand and use. The key in simplifying expressions is to systematically apply rules and calculate step by step.

Positive Exponents

**Positive Exponents** are exponents greater than zero, and they indicate how many times to multiply the base by itself.
Maintaining expressions in positive exponents is often necessary, especially in mathematics and science, where negative exponents might complicate calculations.
For instance, in our exercise, the aim was to express everything using positive exponents. Each step we executed—especially simplifying terms like \(7^0\)—helped achieve this.

• When calculating \(3^4\), the exponent 4 is a positive exponent, indicating multiplication: \(3\) is multiplied by itself four times.
• The calculation \(3 \times 3 \times 3 \times 3 = 81\) clearly illustrates the meaning of positive exponents.

Always aim to express your final answer using positive exponents, as this ensures clarity and standardization in mathematical communication. Positive exponents are generally preferred to maintain uniformity and simplicity.