Problem 33 - Algebra for College Students

Question

Simplify each numerical expression. \(\frac{3^{3}}{3^{-1}}\)

Step-by-Step Solution

Answer

81

Step 1: Understand the Expression

The given expression is \( \frac{3^3}{3^{-1}} \). Our task is to simplify this expression using the properties of exponents.

Step 2: Apply the Quotient Rule for Exponents

The quotient rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Applying this rule, we have \( \frac{3^3}{3^{-1}} = 3^{3 - (-1)} \).

Step 3: Simplify the Exponent

Subtracting the exponents gives us \( 3^{3 + 1} \), since subtracting \(-1\) is the same as adding \(+1\). This simplifies to \( 3^4 \).

Step 4: Calculate the Final Result of the Exponentiation

Finally, calculate \( 3^4 \), which means multiplying \(3\) by itself four times: \(3 \times 3 \times 3 \times 3 = 81\).

Key Concepts

Quotient Rule for Exponents

When you're working with exponents, the quotient rule is a handy tool. This is especially true when you have the same base in both the numerator and denominator. According to the quotient rule,

• if you have a fraction with an exponent in the numerator and an exponent in the denominator, like \( \frac{a^m}{a^n} \), you can simplify by subtracting the exponent in the denominator from the exponent in the numerator: \( a^{m-n} \).
• This rule is only applicable when the bases are the same, such as \( 3^3 \) and \( 3^{-1} \) in our original expression.
• When applying this rule to the example \( \frac{3^3}{3^{-1}} \), you should subtract \(-1\) from \(3\), giving you \(3^{3 - (-1)} = 3^{3 + 1}\).

Using this rule helps you manage and simplify exponential expressions efficiently, making complex math problems easier to handle.

Simplifying Expressions

Simplification in math is all about making expressions easier to work with, without changing their value. When you're simplifying an expression like \( \frac{3^3}{3^{-1}} \), you're aiming to reduce it to its simplest form. This process often involves:

• Applying rules of arithmetic, such as the quotient rule of exponents, to collapse complex parts into simpler, equivalent values.
• Carefully handling operations on exponents. For instance, when you subtract a negative exponent in the denominator from a positive exponent in the numerator, it turns into addition \( 3^{3 + 1} \).
• Once simplified, the expression \( 3^{3+1} \) becomes \( 3^4 \), which is a more straightforward and manageable form.

Breaking down the steps of simplification ensures you understand how each part contributes to the final, simplest form of the expression.

Exponentiation

Exponentiation is the process of raising a number to a power. It's a repeated multiplication of a number by itself. When you see something like \( 3^4 \), it means you multiply \(3\) by itself four times: \(3 \times 3 \times 3 \times 3\).
Here are a few key points about exponentiation:

• The base is the number being multiplied, which is \(3\) in this case.
• The exponent, or power, tells you how many times to multiply the base by itself, which is \(4\).
• Exponentiation is critical in simplifying expressions as it allows you to condense repeated multiplication into a more compact and manageable form.

Calculating \( 3^4 \) gives you \(81\), which is the simplified result of multiplying the base four times. Exponentiation helps model various real-world scenarios like exponential growth and other multiplying processes.