Problem 66 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. $$\frac{2 \cdot(\sqrt{81}-\sqrt[3]{125})}{4^{2}-10+2^{2}}$$

Step-by-Step Solution

Answer

The final calculated value is 0.8.

Step 1: Simplify Square and Cube Roots

Let's start by dealing with each root separately. We have \( \sqrt{81} \) and \( \sqrt[3]{125} \). Calculate these values as follows:\[ \sqrt{81} = 9 \]\[ \sqrt[3]{125} = 5 \]

Step 2: Compute the Numerator

Now substitute the calculated values into the numerator of the expression:\[ 2 \times (\sqrt{81} - \sqrt[3]{125}) = 2 \times (9 - 5) = 2 \times 4 = 8 \]

Step 3: Simplify the Denominator

Next, simplify each part of the denominator:\[ 4^2 = 16 \]Now substitute this into the equation to calculate the total denominator:\[ 16 - 10 + 2^2 = 16 - 10 + 4 = 10 \]

Step 4: Divide the Numerator by the Denominator

Finally, divide the simplified numerator by the simplified denominator:\[ \frac{8}{10} = 0.8 \]

Step 5: Verify with a Calculator

Double-check the calculations using a calculator to ensure accuracy. Input the entire expression and confirm the result is indeed \( 0.8 \).

Key Concepts

Understanding Square Roots

Square roots are all about finding a number that, when multiplied by itself, gives the original number. For instance, the square root of 81 is represented as \( \sqrt{81} \). This means you are looking for a number that when squared equals 81. You'll find that 9 x 9 = 81, which means \( \sqrt{81} = 9 \). It's important to remember:

• Only non-negative numbers have real square roots.
• In our solution, calculating the square root was key to simplifying the numerator.
• Remember to always double-check with a calculator for practice.

Square roots are a fundamental part of algebra, and getting comfortable with them helps in simplifying expressions.

Exploring Cube Roots

Cube roots are similar to square roots but involve cubing. To find the cube root of a number, you find the number that when multiplied by itself three times gives the original value. For example, in \( \sqrt[3]{125} \), we want a number which gives 125 when cubed.

• Here, 5 x 5 x 5 = 125, so \( \sqrt[3]{125} = 5 \).
• Cube roots often appear in problems involving volumes since they are linked to three-dimensions.
• Unlike square roots, cube roots can also be negative since \((-5) \times (-5) \times (-5) = -125\).

In this exercise, understanding cube roots helped simplify the expression further by evaluating \( \sqrt[3]{125} \) to 5.

Simplifying Fractions

Simplifying fractions requires reducing a fraction to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In our example:

• The fraction was \( \frac{8}{10} \).
• The GCD of 8 and 10 is 2.
• Divide both by 2 to get \( \frac{4}{5} \).
• This means \( \frac{8}{10} = 0.8 \), which is its simplest decimal form.

Simplifying fractions makes calculations easier and is especially useful in comparing fractions or expressing fractions in a simpler numerical form.

Basic Arithmetic Operations

Arithmetic operations are the building blocks of math, encompassing addition, subtraction, multiplication, and division. In expressions like the one in our exercise, it's key to:

• Follow the order of operations (PEMDAS/BODMAS), resolving Parentheses, Exponents, Multiplication and Division, then Addition and Subtraction.
• In our numerator, first subtract \( \sqrt[3]{125} \) from \( \sqrt{81} \), then multiply by 2.
• For the denominator, compute each operation in order: powers first, then addition and subtraction.

By following arithmetic rules correctly, you can solve any expression methodically. This ensures precision in your calculations.