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Chapter 4: Problem 233

In the following exercises, simplify. $$\frac{7^{2}+1}{60}$$

Short Answer

Expert verified
The simplified form of \( \frac{7^2 + 1}{60} \) is \( \frac{5}{6} \).

Step by step solution

01

Evaluate the Exponent

First, calculate the value of the exponent in the numerator: a. Determine the value of 7 squared:\( 7^2 = 49 \)b. Add the result to 1:49 + 1 = 50
02

Simplify the Fraction

Next, divide the evaluated numerator by the denominator to simplify the fraction:\( \frac{50}{60} \)
03

Simplify the Fraction Further

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 50 and 60 is 10:\( \frac{50 梅 10}{60 梅 10} = \frac{5}{6} \)

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Key Concepts

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

Evaluating Exponents
When working with exponents, it's important to understand what they represent and how to evaluate them. An exponent tells us how many times a number, called the base, is multiplied by itself. For example, in the expression \(7^2\), 7 is the base and 2 is the exponent. This means that you multiply 7 by itself: \(7 \times 7 = 49\). So, \(7^2 = 49\).

After evaluating the exponent, we proceed by adding any additional numbers in the expression, such as in the given problem: \(7^2 + 1\). Once you find that \(7^2 = 49\), you add 1 to get 50.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is a key concept for simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD is crucial when you need to simplify a fraction fully.

For example, in our problem, we end up with the fraction \( \frac{50}{60} \). To simplify this, we identify the GCD of 50 and 60. Here's how:
  • List the factors of both numbers:
    • Factors of 50: 1, 2, 5, 10, 25, 50
    • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  • Identify the common factors: 1, 2, 5, 10
  • The largest common factor is 10, so 10 is the GCD
Once you have the GCD, you can use it to divide both the numerator and the denominator to simplify the fraction further.
Simplifying Fractions
Simplifying fractions makes them easier to work with and understand. Simplification involves reducing the fraction to its smallest possible numerator and denominator, while keeping the same value.

To simplify a fraction, you divide both the top (numerator) and the bottom (denominator) by their GCD. In our problem, we have the fraction \( \frac{50}{60} \). The GCD is 10, so we divide both the numerator and the denominator by 10 to get: \( \frac{50 \div 10}{60 \div 10} = \frac{5}{6} \). This gives us the simplified fraction.

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Most popular questions from this chapter

In the following exercises, perform the indicated operations and simplify. $$\left(\frac{11}{12}+\frac{3}{8}\right) \div\left(\frac{5}{6}-\frac{1}{10}\right)$$

In the following exercises, add. $$\frac{2}{5}+\frac{1}{5}$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(\frac{m+n}{p}\) when \(m=-6, n=-2, p=4\)

In the following exercises, simplify. $$ \frac{18 r}{27 s} $$

In the following exercises, subtract. $$\frac{11}{15}-\frac{5}{15}-\left(-\frac{2}{15}\right)$$
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