Problem 30 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 3^{\frac{7}{3}} \div 3^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Apply the Property of Exponents

When dividing exponents with the same base, subtract the exponents. Apply this to the expression: \[ 3^{\frac{7}{3}} \div 3^{\frac{1}{3}} = 3^{\left( \frac{7}{3} - \frac{1}{3} \right)} \]

Step 2: Subtract the Exponents

Subtract the exponents: \[ \frac{7}{3} - \frac{1}{3} = \frac{7 - 1}{3} = \frac{6}{3} \]

Step 3: Simplify the Exponent

Simplify the fraction: \[ \frac{6}{3} = 2 \]

Step 4: Express as a Rational Number

Now substitute back the simplified exponent into the base expression: \[ 3^2 \] Simplify this to get: \[ 3^2 = 9 \]

Key Concepts

Understanding Properties of Exponents

The properties of exponents make manipulating and simplifying expressions with exponents much more manageable. One of the key properties is the 'quotient of powers' property, which tells us that when we divide exponential expressions with the same base, we subtract their exponents. For example, in the expression \(a^m \div a^n\), we can simplify it to \(a^{m-n}\). Other essential properties include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^m = a^m \times b^m\)
• Zero Exponent: \(a^0 = 1\) if \(a eq 0\)

These rules are useful when simplifying complex algebraic expressions involving exponents.

Base and Exponents Clarified

In mathematics, a base and an exponent are used to express repeated multiplication. The base is the number that gets multiplied, and the exponent tells us how many times to multiply the base by itself. For example, in \(3^4\), 3 is the base, and 4 is the exponent, meaning you should multiply 3 by itself four times: \(3 \times 3 \times 3 \times 3\).
This concept is key when expressing numbers in powers, enabling a compact representation and simplifying calculation, especially when combining terms with similar bases.

Doing Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, the fraction \( \frac{6}{3} \) can be simplified by dividing both terms by 3, resulting in \( \frac{2}{1} \) or simply 2.
Simplifying fractions streamlines computation and makes it easier to identify equivalent proportions and relationships in mathematical expressions. Pay attention to such simplifications when working with exponents and rational numbers to solve problems efficiently.

Applying Algebra 2 Solutions Strategies

Algebra 2 problems often require a strategic approach to simplify expressions and solve equations. Key strategies include mastering exponent rules, which empowers us to tackle expressions with the same base efficiently.

• Identify like terms and similar bases, so you can apply exponent rules.
• Look for opportunities to simplify parts individually, like simplifying fractions.
• Double-check by substituting back simplified values to verify your solution.

Being consistent and aware of these strategies can lead smoothly from understanding a problem to applying the right algebraic rules, eventually arriving at simplified and correct solutions.