Problem 13 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ \left(5^{101}\right)^{1 / 101} $$

Step-by-Step Solution

Answer

5

Step 1: Understand the Expression

The expression given is \( \left(5^{101}\right)^{1 / 101} \). This is an example of applying an exponent on a base number followed by taking the root of the power.

Step 2: Apply Exponentiation Rules

We use the exponentiation rule that states: \( (a^m)^n = a^{m \cdot n} \). Here, \( a = 5 \), \( m = 101 \), and \( n = \frac{1}{101} \). Substitute these values into the rule.

Step 3: Simplify the Exponents

Perform the multiplication inside the exponent: \( 5^{101 \cdot \frac{1}{101}} \). This simplifies to \( 5^{101/101} \).

Step 4: Simplify Further

Since \( 101/101 = 1 \), the exponent simplifies further to \( 5^1 \).

Step 5: Final Result

The expression \( \left(5^{101}\right)^{1 / 101} \) simplifies to \( 5 \).

Key Concepts

Exponent Rules

Exponent rules are fundamental guidelines that help in simplifying expressions involving powers. These rules make working with large numbers easier by breaking down the task into manageable parts. Here are some basic exponent rules:

• Product of Powers Rule: If you multiply two exponents with the same base, you add their powers. For example, \( a^m \times a^n = a^{m+n} \).
• Power of a Power Rule: When raising an exponent to another power, you multiply the exponents. This rule was used in the original exercise: \( (a^m)^n = a^{m \cdot n} \).
• Power of a Product Rule: If you raise a product to a power, each factor in the product gets raised to that power. For example, \( (ab)^m = a^m b^m \).
• Zero Exponent Rule: Any base raised to the power of zero is one, \( a^0 = 1 \), where \( a eq 0 \).
• Negative Exponent Rule: A negative exponent indicates a reciprocal, \( a^{-m} = \frac{1}{a^m} \).
• Quotient of Powers Rule: If you divide two exponents with the same base, you subtract their powers, \( \frac{a^m}{a^n} = a^{m-n} \).

Understanding these rules allows for the simplification of even the most complex expressions. They serve as the building blocks in algebra, making expressions easier to manage and solve. As seen in the solution, applying the Power of a Power Rule, we simplified \( \left(5^{101}\right)^{1/101} \) to a basic power.

Simplifying Expressions

Simplifying expressions in algebra means converting them into the simplest form possible. It involves breaking down complex expressions using mathematical rules and reducing them to a form that is easier to understand and work with.

• Combine like terms: Terms that share the same variable and power can be added or subtracted. For example, \( 2x + 3x = 5x \).
• Use the distributive property: This allows the multiplication of terms within parentheses to be simplified. For example, \( a(b + c) = ab + ac \).
• Apply exponent rules: These help expand or compress powers in expressions, as seen in the original exercise.
• Factor where possible: Expressions can be made simpler by factoring common terms. For example, \( x^2 + 5x = x(x + 5) \).

In the original exercise, the expression \( \left(5^{101}\right)^{1/101} \) was simplified to \( 5 \) using exponent rules. The step-by-step reduction created a simpler form that’s easy to interpret and use.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation symbols. They are like phrases that describe mathematics without forming a complete equation or inequality. Understanding and manipulating these expressions is key in solving algebraic problems.

• Variables: Symbols that stand for unknown values, usually represented by letters like \( x \), \( y \), or \( z \).
• Constants: Known numbers that aren’t prone to change, like \( 3 \), \( -7 \), or \( \pi \).
• Coefficients: Numbers that multiply a variable within the expression, as in \( 4x \) where \( 4 \) is the coefficient.
• Operators: Symbols that denote actions within expressions such as addition \( + \), subtraction \( - \), multiplication \( \times \), and division \( \div \).

In the original exercise, \( \left(5^{101}\right)^{1/101} \), the primary focus was on the correct application of exponent rules to simplify the expression. Recognizing the pattern of variables and constants helps in understanding the structure of such algebraic expressions. Grasping how to manipulate these elements using algebraic rules is vital for anyone studying algebra.