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Chapter 1: Problem 401

Simplify. $$ \frac{4^{2}-1}{25} $$

Short Answer

Expert verified
The simplified form is \( \frac{3}{5} \).

Step by step solution

01

Evaluate the Numerator

First, simplify the numerator, which is given by an expression involving exponents: \(4^{2} - 1\). Calculate \(4^{2}\) which equals 16. So, the numerator becomes 16 - 1.
02

Simplify the Numerator's Result

Next, subtract 1 from 16: \(16 - 1 = 15\). Hence, the numerator simplifies to 15.
03

Simplify the Fraction

Now, place the simplified numerator back in the fraction: \(\frac{15}{25}\). Simplify the fraction by finding the greatest common divisor of 15 and 25 which is 5. Divide both the numerator and the denominator by 5.\( \frac{15 \div 5}{25 \div 5} = \frac{3}{5} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

numerator
In any fraction, the numerator is the top number. It tells you how many parts of the whole you have. For example, in the fraction \(\frac{4}{5}\), 4 is the numerator. The numerator can be a simple number or a more complex expression that you might need to simplify. In our problem, the numerator was initially given as \(4^{2} - 1\). Simplifying this step by step helps in making the overall simplification easier.

Remember, the numerator and the denominator work together to define the value of the fraction. The larger the numerator compared to the denominator, the larger the value of the fraction will be.
denominator
The denominator is the bottom number in a fraction. It tells you how many equal parts the whole is divided into. For instance, in the fraction \(\frac{4}{5}\), 5 is the denominator.

The denominator helps to understand the scale or size of the pieces we are dealing with. A larger denominator means smaller pieces and vice versa. In our fraction \(\frac{4^{2}-1}{25}\), 25 is the denominator. The denominator remains unchanged during the simplification of the numerator, and it is important as you divide both the numerator and the denominator by their greatest common divisor.
exponents
Exponents are used to show how many times a number (called the base) is multiplied by itself. For example, in \(4^{2}\), 4 is the base and 2 is the exponent. It means 4 multiplied by itself, which equals 16.

In our problem, the exponent is key to simplifying the numerator. We first calculated \(4^{2}\), which equals 16, then subtracted 1 to find that the numerator simplifies to 15. Understanding how to work with exponents can simplify complex expressions and make it easier to manage larger numbers.
greatest common divisor
The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is essential in simplifying fractions. In the fraction \(\frac{15}{25}\), the GCD of 15 and 25 is 5.

To simplify the fraction, we divide both the numerator (15) and the denominator (25) by the GCD. This results in \(\frac{3}{5}\), which is the simplified form of the fraction. Knowing how to find the GCD helps to ensure fractions are reduced to their simplest form.

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