Problem 60 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(\frac{6^{2}-1}{2^{3}-3}+\frac{4^{3}+2 \cdot 3}{2 \cdot 5}\)

Step-by-Step Solution

Answer

The value is 14.

Step 1: Calculate the first expression's numerator

The first expression in the given formula is \( \frac{6^2 - 1}{2^3 - 3} \). First, compute \( 6^2\). Since \( 6^2 = 36 \), then the numerator is \( 36 - 1 = 35 \).

Step 2: Calculate the first expression's denominator

For the first expression's denominator, \(2^3\) must be calculated. \(2^3 = 8\), so the denominator becomes \(8 - 3 = 5 \).

Step 3: Simplify the first expression

Now we can simplify \( \frac{6^2 - 1}{2^3 - 3} \) to \( \frac{35}{5} = 7 \).

Step 4: Calculate the second expression's numerator

The second expression in the equation is \(\frac{4^3 + 2 \cdot 3}{2 \cdot 5} \). First, compute the term \(4^3\), which results in \(64\). Then, \(2 \cdot 3 = 6\), so the entire numerator is \(64 + 6 = 70\).

Step 5: Calculate the second expression's denominator

For the second expression's denominator, calculate \(2 \cdot 5\), which equals \(10\).

Step 6: Simplify the second expression

Simplify \(\frac{4^3 + 2 \cdot 3}{2 \cdot 5}\) to \(\frac{70}{10} = 7\).

Step 7: Add the results of the simplified expressions

Add the two results obtained: \(7 + 7 = 14\). Therefore, the final result is \(14\).

Key Concepts

Exponents

Exponents are a shorthand way to represent repeated multiplication of the same number. When you see an expression like \(6^2\), it means you multiply 6 by itself: 6 times 6. This results in 36. The exponent (in this case 2) tells us how many times the base (6) is used as a factor. This concept is essential for simplifying arithmetic expressions, which is exactly what we've seen in the problem. In our initial exercise, we used exponents to find the correct values for both \(6^2\) and \(2^3\).
Pay close attention to the placement of the exponent. A higher exponent increases the result significantly, as seen in \(4^3\), which equals 64 because 4 is multiplied by itself three times. This might seem challenging at first, but with practice, using exponents can make calculations much quicker.

Numerator and Denominator

In a fraction, the top part is called the numerator, and the bottom part is called the denominator. Understanding these terms is crucial when working with fractions. They help you determine the value of that fraction.
Let's take the original expression \( \frac{6^2 - 1}{2^3 - 3} \). Here, '6 squared minus 1' (which is 35) is the numerator. The denominator '2 cubed minus 3' results in 5.

• The numerator tells us how many parts of a whole we have.
• The denominator indicates how many equal parts the whole is divided into.

Having a good understanding of these components enables us to perform operations like simplifying fractions, as each step becomes more intuitive once you grasp what each part represents.

Simplifying Fractions

Simplifying fractions is an invaluable skill, especially when it comes to evaluating expressions and solving equations. Simplifying makes fractions easier to interpret and can reveal their simplest form, where the numerator and the denominator have no common factors except 1. This means the fraction can't be reduced further.
In the problem we solved, the fraction \( \frac{35}{5} \) was simplified to 7, because 35 divided by 5 results in a whole number. Similarly, \( \frac{70}{10} \) simplifies to 7. Simplifying involves finding a common factor of the numerator and denominator and dividing both by this factor.

• The process often begins by factoring both parts of the fraction.
• Then, you divide them by their greatest common divisor (GCD).

Mastering this process helps you to recognize and work more quickly with simpler expressions, as well as ensuring that your results are clear and concise.