Problem 5 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 25^{3 / 2} $$

Step-by-Step Solution

Answer

The expression \(25^{3 / 2}\) simplifies to 125.

Step 1: Analyze the exponent

The problem is to evaluate \(25^{3 / 2}\). Look at the exponent. It's fractional, meaning it carries the operation of taking both power and root. The top number in the fraction indicates the power to raise numbers to, and the bottom number indicates the root to take.

Step 2: Apply the exponent

Apply the power of 3 to the base number 25. This means multiplying 25 by itself three times: \(25*25*25 = 15625\).

Step 3: Apply the root

The bottom number of the fractional exponent indicates the type of root to take. In this case, it's 2, which means take a square root. The square root of 15625 is 125. So, \(25^{3 / 2} = 125\).

Step 4: Verification (Optional)

To check that \(25^{3 / 2} = 125\), we can use the property that \( (a^{b/c})^c = a^b \) to get \( (125^2) = 15625 = 25^3 \), hence confirming our solution.

Key Concepts

Understanding Exponents

Exponents are a fundamental mathematical concept that allows you to express repeated multiplication in a concise way. If you see an expression like \( a^n \), this means you multiply the base \( a \) by itself \( n \) times. Here's how it works:

• The number or expression being multiplied is called the "base." In \( a^n \), the base is \( a \).
• The "exponent" \( n \), tells you how many times you use the base in a multiplication.

For example, \( 5^3 \) means \( 5 \times 5 \times 5 = 125 \). It’s important to remember:

• If \( n = 0 \), then \( a^0 = 1 \) for any \( a eq 0 \).
• Negative exponents \( a^{-n} \) represent fractions: \( \frac{1}{a^n} \).
• An exponent of 1 means the number itself: \( a^1 = a \).

Understanding these basics is crucial when working with any kind of exponents, including fractional ones.

Exploring Fractional Exponents

Fractional exponents might seem a bit tricky at first, but they link directly to roots. A fractional exponent like \( a^{b/c} \) tells you to both take a root and apply a power.

• The denominator \( c \) of the fraction indicates the root you should take: the \( c \)-th root of the base.
• The numerator \( b \) indicates the power you should raise the base to.

To evaluate \( a^{b/c} \):

1. Take the \( c \)-th root of \( a \).
2. Raise the result to the \( b \) power.

For \( 25^{3/2} \):1. We notice the fraction \( \frac{3}{2} \) tells us two things: take the square root of 25 and then cube the result. 2. First, compute the square root of 25, which is 5.3. Then, raise 5 to the power of 3: \( 5^3 = 5 \times 5 \times 5 = 125 \).Understanding fractional exponents is valuable, as they simplify the process of working with numbers in different dimensions.

Demystifying Square Roots

Square roots are one of the most commonly encountered roots in mathematics. Finding the square root of a number means identifying a value that, when multiplied by itself, equals that number. The square root of a number \( x \) is denoted as \( \sqrt{x} \).A few key points to recall when dealing with square roots:

• The most straightforward example is \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).
• Square roots always come in pairs—positive and negative, because both \( 3 \times 3 = 9 \) and \( -3 \times -3 = 9 \). However, when you see \( \sqrt{x} \), it usually refers to the principal, or positive, square root.
• Numbers not perfect squares (like 10) have square roots that can be approximations (non-integers). For example, \( \sqrt{10} \approx 3.162 \).

In the exercise of evaluating \( 25^{3/2} \), the square root part corresponds to the denominator 2 in the fractional exponent. Therefore, finding the square root is an essential step before applying the exponent in any calculation involving fractional powers.