Problem 52 - 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. $$ \frac{4^{16}}{4^{13}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{4^{16}}{4^{13}}.$$ Answer: 64

Step 1: Recall the quotient rule of exponents

The quotient rule of exponents states that, when we divide two powers with the same base, we subtract the exponent of the denominator from the exponent of the numerator: $$\frac{a^m}{a^n} = a^{(m-n)}.$$

Step 2: Apply the quotient rule to the given expression

We have the expression $$\frac{4^{16}}{4^{13}}.$$ Using the quotient rule of exponents, we subtract the exponent of the denominator (13) from the exponent of the numerator (16): $$\frac{4^{16}}{4^{13}} = 4^{(16-13)}.$$

Step 3: Simplify the expression

Now, subtract the exponents: $$4^{(16-13)} = 4^3.$$

Step 4: Compute the final result

Finally, calculate the result of the power: $$4^3 = 4 \times 4 \times 4 = 64.$$ So, $$\frac{4^{16}}{4^{13}} = 64.$$

Key Concepts

product rule of exponents

The product rule of exponents is a helpful tool when you are multiplying expressions that have the same base. It simplifies calculations and makes it easier to work with exponents neatly. The rule states:

• For any nonzero base \(a\) and whole numbers \(m\) and \(n\), \(a^m \times a^n = a^{m+n}\).

This means you simply add the exponents together. For example, if you encounter \(x^3 \times x^4\), you just add 3 and 4 to get \(x^{7}\).
This rule only applies when the bases are the same. You cannot apply this rule to expressions with different bases, such as \(2^3 \times 3^3\).
Using the product rule can significantly streamline the process of simplifying expressions, especially when dealing with large exponents. It is essential to ensure the bases match before applying this rule, as misapplying it could lead to incorrect results. Always double-check your base before proceeding!

simplifying expressions

Simplifying expressions is a crucial aspect of algebra and helps in breaking down complex problems into more manageable parts.
When simplifying, your goal is to make the expression as straightforward as possible without changing its value. Using rules such as the quotient rule of exponents, which was outlined in the original step-by-step solution, is one method to achieve this.
Here's a brief reminder of what the quotient rule entails:

• You take two powers with the same base and subtract the exponent in the denominator from the exponent in the numerator, thus \(\frac{a^m}{a^n} = a^{(m-n)}\).

This results in a **simpler expression** that is easier to compute.
In our given example, \(\frac{4^{16}}{4^{13}}\), this rule allows us to reduce the expression to \(4^3\), which is simpler and quicker to evaluate.
Achieving a clear, simplified expression not only verifies your mathematical process but also helps in solving more complex equations later on.

exponents calculation

Calculating exponents is a vital skill, often necessary for solving various mathematical problems.
An exponent indicates how many times a number, known as the base, is multiplied by itself. In other words, \(a^3\) means \(a \times a \times a\).
When working with exponents, it’s important to follow specific rules and methods to ensure accurate results.

• The **quotient rule of exponents** helps you divide powers of the same base, simplifying your calculations by subtracting exponents.
• The **product rule of exponents** allows you to multiply powers while just adding the exponents, providing an easy way to condense expressions.

For example, with our exercise, the sequence of calculating \(4^3\) involves multiplying 4 by itself three times:
\(4 \times 4 = 16\), and then \(16 \times 4 = 64\). This simple calculation illustrates the power of understanding and applying exponent rules correctly.
Accurate exponent calculation is not just about obtaining the correct answer but also about deepening your understanding of mathematical relationships and improving your problem-solving skills.