Problem 17 - College Algebra

Question

Evaluate each exponential expression. $$ \frac{2^{8}}{2^{4}} $$

Step-by-Step Solution

Answer

The result of evaluating the expression \(\frac{2^{8}}{2^{4}}\) is 16.

Step 1: Identify the base and Exponents

In this exponential expression, our base is 2 and the exponents are 8 and 4 respectively. The equation is given as \(\frac{2^{8}}{2^{4}}\)

Step 2: Use the Laws of Exponents

We will use the quotient of powers property of exponents which states that for any non-zero number a, and integers m and n: \(a^{m}/a^{n}=a^{m-n}\). So, we subtract the exponent in the denominator from the exponent in the numerator.

Step 3: Subtract the Exponents

We subtract the exponent 4 from 8: \(2^{8-4}\).

Step 4: Simplify the Exponents

Which gives us \(2^{4}\).

Step 5: Evaluate the Result

Finally, we evaluate \(2^{4}\) which equals to 16.

Key Concepts

Laws of Exponents

Exponential expressions often involve working with laws of exponents that help simplify the calculations. One of the main laws is the Product of Powers, which allows you to multiply two exponents with the same base by adding their powers. For instance, if you have \[ a^m \times a^n = a^{m+n} \]Another crucial law is the Quotient of Powers, which we'll explore further in a moment, but it essentially simplifies division of like bases by subtracting the exponents. There's also the Power of a Power rule, which involves multiplying the exponents when a power is raised to another power:\[ (a^m)^n = a^{m \cdot n} \]Understanding these laws can be incredibly helpful when simplifying or rearranging exponential expressions.

Quotient of Powers Property

The Quotient of Powers Property is a rule of exponents that says you can divide two powers with the same base by subtracting their exponents. The formula looks like this:\[ \frac{a^m}{a^n} = a^{m-n} \]When the base is the same, simply take the exponent from the denominator and subtract it from the exponent in the numerator.

Examples

If you have \( \frac{3^5}{3^3} \), you subtract the exponents: \[ 3^{5-3} = 3^2 \]The answer is around 9, since \(3^2 = 9\). Knowing this property allows you to quickly evaluate large or complex exponential expressions by reducing them to simpler forms.

Evaluate Exponential Expressions

Evaluating exponential expressions means finding their numerical value. Start by identifying the base and the exponent. An expression like \(2^4\) indicates that 2 is the base and 4 is the exponent, meaning you multiply the base, 2, by itself 4 times: \[ 2 \times 2 \times 2 \times 2 = 16 \]For expressions that use the quotient of powers property, like \(\frac{2^8}{2^4}\), first apply the property by subtracting exponents, then calculate: \[ 2^{8-4} = 2^4 = 16 \]This process simplifies potentially complicated expressions into simpler calculations.

Simplify Exponents

Simplifying exponents is about making expressions as simple as possible. You do this by using exponent rules to rearrange and reduce terms.

Why Simplify?

To make calculations easier and results more understandable. A simpler form is quicker to evaluate.Start by applying laws like the quotient or product of powers, and then reduce to a single term if possible. For instance, \[ (2^3)^2 \] simplifies by using the power of a power rule, becoming \[ 2^{3 \cdot 2} = 2^6 \]By implementing these methods effectively, you can transform complex expressions into straightforward numbers or simpler exponentials, enhancing mathematical efficiency.