Problem 52 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left(4^{3}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4^{9} \).

Step 1: Identify the Power Rule

The power rule for exponents states that when you raise a power to a power, you multiply the exponents. This can be expressed as \( (a^m)^n = a^{m \cdot n} \). We will use this rule to simplify the expression \( (4^{3})^{3} \).

Step 2: Apply the Power Rule

Using the power rule, we calculate the new exponent by multiplying the existing exponents: \( 3 \times 3 = 9 \). This means that \( (4^{3})^{3} \) becomes \( 4^{9} \).

Step 3: Express in Simplified Form

Now that we have applied the multiplication of exponents, the simplified expression is \( 4^{9} \). The expression no longer needs to be computed further, as the problem asks for it in exponential form.

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we mean making them easier to understand or work with. In the context of exponents, simplifying often involves reducing a complex expression to a more straightforward form using the rules of exponents.
This doesn't necessarily mean solving the expression entirely; rather, it's about rewriting it in a simpler manner.
For the expression \( (4^{3})^{3} \), simplifying it involves applying the power rule for exponents. This rule helps us tidy up our work by minimizing the number of operations needed.

• The goal is to make the expression as compact as possible, so it's more manageable in further calculations or comparisons.

In this case, we don't need to calculate the exact value of \( 4^{9} \) because presenting it in exponential form is both appropriate and efficient.

Multiplying Exponents

Multiplying exponents is an operation that occurs when we apply the power rule, which states that to find the power of a power, we multiply the exponents. This simplifies the process of working with expressions that involve repeated powers.
Consider the expression \( (a^m)^n \). According to the power rule, this becomes \( a^{m \cdot n} \). This means, rather than dealing with multiple sequential multiplication steps, you can get your answer by simply multiplying the exponents.

• For \( (4^3)^3 \), you multiply the exponents: \( 3 \times 3 = 9 \), hence the expression simplifies to \( 4^9 \).
• This rule drastically cuts down on the work needed to simplify nested exponentials, converting them into a single power of the base.

Using this concept effectively allows us to handle larger and more complex problems with ease.

Exponential Form

Exponential form is a mathematical notation that involves expressing numbers using a base and an exponent. This format not only represents numbers succinctly but also makes computation easier, especially when dealing with powers and roots.
In \( 4^{9} \), for instance, the number 4 is the base, and 9 is the exponent, indicating that 4 should be multiplied by itself 9 times.

• Exponents count how many times the base is used as a factor.
• In the case of \( 4^{9} \), it is more efficient to leave the expression in this form rather than expanding it into a lengthy multiplication.

Using exponential form is invaluable for simplifying and handling very large numbers or expressions in mathematics without getting bogged down in extensive arithmetic.