Problem 32 - Algebra 2 and Trigonometry

Question

In \(3-37,\) express each power as a rational number in simplest form. $$ 125^{\frac{2}{3}} \div 125^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplest form of the expression is 5.

Step 1: Understanding the Expression

The expression given is \(125^{\frac{2}{3}} \div 125^{\frac{1}{3}}\). This means we need to divide the two powers of 125. When dividing two numbers with the same base, we subtract their exponents.

Step 2: Apply the Division of Exponents Rule

According to the rule, \(a^m \div a^n = a^{m-n}\). Here, the base \(a\) is 125, \(m = \frac{2}{3}\), and \(n = \frac{1}{3}\). Substitute these into the division of powers formula: \(125^{\frac{2}{3}} \div 125^{\frac{1}{3}} = 125^{\frac{2}{3} - \frac{1}{3}}\).

Step 3: Simplify the Exponents

Calculate \(\frac{2}{3} - \frac{1}{3}\). This simplifies to \(\frac{1}{3}\). So, the expression reduces to \(125^{\frac{1}{3}}\).

Step 4: Evaluate the Power

Now evaluate \(125^{\frac{1}{3}}\). This is the cube root of 125. Since \(5^3 = 125\), the cube root of 125 is 5.

Key Concepts

Exponent Rules

Exponent rules help us manage expressions involving powers and exponents more easily. There are several rules to remember, but the one we are focusing on here is for dividing exponents. When you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator:

• If you have an expression like \( a^m \div a^n \), it simplifies to \( a^{m-n} \).

Dividing exponents rule is very useful when simplifying expressions with similar bases, as it allows you to combine terms effectively. Just remember to always check for the same base and use subtraction to find the resulting power. This rule can make complex algebraic expressions much easier to handle. To apply this rule correctly, be careful with negative and fractional exponents, as they can alter the behavior of the expression significantly.

Cube Roots

A cube root answers the question: what number, when multiplied by itself twice, gives us the original number? In mathematical terms, the cube root of a number \( a \) is another number \( b \) such that \( b^3 = a \).

• Example: The cube root of 125, denoted as \( 125^{\frac{1}{3}} \), is 5 because \( 5^3 = 125 \).

Understanding cube roots is vital in simplifying expressions that contain rational exponents. When you encounter a rational exponent like \( \frac{1}{3} \), it indicates you should find the cube root. Finding cube roots often involves recognizing perfect cubes or using a calculator for more complex numbers. Cube roots are one of the essential concepts in algebra and can simplify or solve real-world problems effectively.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with. The goal is to reduce an expression to its simplest form, which often involves combining like terms, using exponent rules, and performing arithmetic operations.

• Start by looking for common bases and apply the rules of exponents.
• Make use of roots and rational exponents to further simplify the expression.

In our example, \( 125^{\frac{2}{3}} \div 125^{\frac{1}{3}} \), the simplification begins with using the division rule for exponents to combine terms with the same base. This results in \( 125^{\frac{1}{3}} \), which is then simplified by evaluating the cube root. Simplifying expressions not only makes complex problems easier to solve but also helps in verifying solutions and ensuring they are accurate. Keeping expressions in simple form saves time and reduces errors in computations.