Problem 10 - Algebra and Trigonometry

Question

Evaluate each expression. $$ 5^{2} \cdot\left(\frac{1}{5}\right)^{3} $$

Step-by-Step Solution

Answer

\(\frac{1}{5}\)

Step 1: Evaluate the Power of 5

First, calculate the value of the expression for the power of 5. \(5^2 = 5 \times 5 = 25\).

Step 2: Evaluate the Power of Fraction of 5

Next, calculate \(\left(\frac{1}{5}\right)^3\). This means multiplying the fraction \(\frac{1}{5}\) by itself three times: \(\left(\frac{1}{5}\right)^3 = \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} = \frac{1}{125}\).

Step 3: Multiply the Results

Finally, multiply the results from Step 1 and Step 2. Multiply 25 by \(\frac{1}{125}\):\[ 25 \times \frac{1}{125} = \frac{25}{125} = \frac{1}{5} \].

Key Concepts

Powers of Numbers

In mathematics, powers of numbers, also known as exponents, tell us how many times to use the number in a multiplication. For example, in the expression \(5^2\), the number 5 is known as the base, and 2 is the exponent. This means you multiply the base, 5, by itself: \(5 \times 5\). Therefore, \(5^2 = 25\).

Understanding powers is crucial because they show repeated multiplication simply and compactly.

• When the exponent is 2, it's often referred to as "squared" because it represents the area of a square with sides of that length.
• When the exponent is 3, it's called "cubed," symbolizing the volume of a cube with edges of that length.

Knowing how to calculate powers swiftly and accurately enhances problem-solving skills, especially in algebra and higher mathematics.

Multiplying Fractions

Multiplying fractions might seem daunting, but it's actually straightforward with practice. The key rule is to multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, if you need to multiply \(\frac{1}{5}\) by itself three times, you'll follow this process:

\[\left(\frac{1}{5}\right)^3 = \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} = \frac{1 \times 1 \times 1}{5 \times 5 \times 5} = \frac{1}{125}\]

Working with fractions often involves:

• Finding a common denominator is unnecessary when multiplying.
• Simplifying fractions as the final step, if possible, to maintain simplicity in your results.

Practicing with fractions builds a strong foundation for more complex math problems involving fractions and rational numbers.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest possible version, where the numerator and denominator have no common factors other than 1. It makes the fraction easier to understand and work with, especially when comparing them.

For example, after multiplying the fractions from the previous steps, we get \(\frac{25}{125}\). To simplify this fraction, follow these steps:

1. **Find the Greatest Common Divisor (GCD):** The GCD of 25 and 125 is 25.
2. **Divide both the numerator and denominator by the GCD:**
\[\frac{25}{125} = \frac{25 \div 25}{125 \div 25} = \frac{1}{5}\]

Simplifying ensures your solution is precise and the most elegant form:

• Always check each factor to ensure there's no common divisor left.
• Fractions become more intuitive once simplified, aiding clarity in solving math problems.

Mastering simplification aids in efficiently solving mathematical equations and enhances overall number sense.