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Chapter 4: Problem 232

In the following exercises, simplify. $$\frac{4^{2}-1}{25}$$

Short Answer

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

Step by step solution

01

Identify the Expression

The given expression to simplify is \( \frac{4^{2}-1}{25} \).
02

Calculate the Exponent

First, calculate \( 4^{2} \). \( 4^{2} = 16 \).
03

Substitute and Simplify the Numerator

Substitute the value in the numerator: \( 16 - 1 \). Simplify it to get \( 15 \).
04

Simplify the Fraction

The fraction now is \( \frac{15}{25} \). Simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5. \( \frac{15 \text{ 梅 5}}{25 \text{ 梅 5}} = \frac{3}{5} \).

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Key Concepts

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

Simplifying expressions
When working with algebraic expressions, simplifying involves reducing them to their simplest form without changing their values. Here, the goal is to perform operations like addition, subtraction, multiplication, and division to achieve a simpler expression. Simplifying fractions is one frequent task in this process. For example, we started with the fraction \( \frac{4^{2} - 1}{25} \). First, we need to deal with the exponent in the numerator. \
Next, we simplify each part of the numerator, and lastly, we break down the fraction using our understanding of the greatest common divisor. The basics steps found in the exercise are essential for mastering algebra.
Exponents
Exponents are a shorthand method of indicating repeated multiplication of a number by itself. For example, \( 4^{2} \) means 4 multiplied by 4. The number 4 is the base, and 2 is the exponent. In the example, calculating \( 4^{2} = 16 \) was the first step.
Understanding how to work with exponents is crucial for simplifying expressions.
Always resolve the exponent first before any other operations in the expression. This is a part of the order of operations (PEMDAS/BODMAS), where you handle Parentheses/Brackets first, then Exponents/Orders.
Greatest common divisor
The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest integer that divides two or more integers without leaving a remainder. For example, the GCD of 15 and 25 is 5.
Simplifying a fraction involves dividing both the numerator and denominator by their GCD to get the simplest form of the fraction.
In our exercise, after simplifying the numerator to 15, the fraction was \( \frac{15}{25} \). We determined the GCD of 15 and 25 is 5.
When both parts of the fraction are divided by 5, we get \( \frac{3}{5} \). Knowing how to find the GCD is essential for simplifying fractions and ensuring they are in their simplest forms.

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Most popular questions from this chapter

In the following exercises, perform the indicated operations and simplify. $$\frac{2}{5}+\left(-\frac{5}{9}\right)$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{r-s}{r+s}\) when \(r=10, s=-5\)

In the following exercises, find the least common denominator. $$\frac{1}{3} \text { and } \frac{1}{12}$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{5}{12}-w\) when (a) \(w=\frac{1}{4} \quad\) (b) \(w=-\frac{1}{4}\)

In the following exercises, determine whether the each number is a solution of the given equation. $$ y+\frac{3}{5}=\frac{5}{9} $$ $$ (a)y=\frac{1}{2} $$ $$ (b)y=\frac{52}{45} $$ $$ (c)y=-\frac{2}{45} $$
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