Problem 14 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$25^{-3 / 2}$$

Step-by-Step Solution

Answer

\(\frac{1}{125}\)

Step 1: Understanding the Expression

The expression given is \(25^{-3/2}\). It has a base of 25 raised to the power of \(-3/2\). Negative exponents indicate a reciprocal, and a fractional exponent represents a root and a power.

Step 2: Convert the Negative Exponent

Begin by converting the negative exponent into a positive one using the reciprocal. This turns \(25^{-3/2}\) into \(\frac{1}{25^{3/2}}\).

Step 3: Simplify the Fractional Exponent

The fractional exponent \(3/2\) can be split into two parts: the root (denominator) and the power (numerator). First, find the square root since the denominator is 2.

Step 4: Calculate the Square Root

Find the square root of 25. Since \(\sqrt{25} = 5\), substitute this into the expression, which becomes \(\frac{1}{5^3}\).

Step 5: Apply the Power

Now, apply the power of 3 to 5. Calculate \(5^3\), which equals 125.

Step 6: Calculate the Final Expression

The expression has simplified to \(\frac{1}{125}\), so \(25^{-3/2} = \frac{1}{125}\).

Key Concepts

Exponent Rules

Exponent rules are fundamental to understanding how to manipulate expressions involving powers. Exponents indicate how many times a base is multiplied by itself.
There are key rules that help simplify these expressions:

• The Product of Powers Rule: When multiplying numbers with the same base, add the exponents. For example,
  \( x^a \times x^b = x^{a+b} \).


• The Power of a Power Rule: In this scenario, multiply the exponents, such as \((x^a)^b = x^{a \cdot b} \).


• The Quotient of Powers Rule: When dividing numbers with the same base, subtract the exponents:
  \( \frac{x^a}{x^b} = x^{a-b} \).


• Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the positive power:
  \( x^{-a} = \frac{1}{x^a} \).

Understanding these rules makes it easier to grasp other concepts associated with exponents, such as fractional exponents.

Fractional Exponents

Fractional exponents can seem tricky at first, but they are simply another way to represent powers and roots. A fractional exponent allows for more flexibility in expressing numbers.
Here's how they work:

• The numerator of the fraction represents the power: For example, in \( x^{m/n} \), \( m \) is the power.


• The denominator denotes the root: Continuing with \( x^{m/n} \), \( n \) is the root.
  Thus, \( x^{m/n} = \sqrt[n]{x^m} \).


• If the denominator is 2, it implies a square root: For instance, \( x^{1/2} \) is the square root of \( x \), or \( \sqrt{x} \).


• The base is subjected to the root first, then the result is raised to the power: In the example \( 27^{2/3} \), you first find the cube root of 27, and then square the result.

By breaking down fractional exponents, it becomes clear how to handle complex-looking expressions with ease.

Powers and Roots

Powers and roots are closely related concepts in mathematics, and understanding their relationship is essential.
Here is a brief breakdown:

• Powers (or exponents) indicate how many times to multiply the base by itself: For example, \( 3^4 \) means multiply 3 by itself 4 times \((3 \times 3 \times 3 \times 3 = 81)\).


• Roots are the opposite of powers: Where a power is a repeated multiplication, a root "undoes" that process.
  For instance, the square root \( \sqrt{81} = 9 \), because \( 9^2 = 81 \).
• Roots can also be represented using exponents: For example, the square root of \( x \) is the same as \( x^{1/2} \).


• Both concepts are reversible: Just as you can calculate powers, you can also determine roots, creating a balanced understanding of powers and roots.

By getting comfortable with these concepts and their interactions, one can simplify and solve expressions that incorporate both powers and roots effectively.