Problem 76 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ \left(\frac{9+2 \cdot 0}{9}\right)^{2} $$

Step-by-Step Solution

Answer

1

Step 1: Understand the expression

The given expression to evaluate is \[ \left(\frac{9 + 2 \cdot 0}{9}\right)^{2} \]. We'll break it down into smaller parts and solve it step-by-step.

Step 2: Simplify the numerator

Simplify the numerator by performing the operations inside the parentheses. The numerator is \[ 9 + 2 \cdot 0 \]. Since anything multiplied by 0 is 0, we get \[ 9 + 0 = 9 \].

Step 3: Divide the numerator by the denominator

Now divide the simplified numerator by the denominator: \[ \frac{9}{9} = 1 \].

Step 4: Square the result

Finally, take the result and square it: \[ 1^{2} = 1 \].

Key Concepts

numerator simplification

In this section, we'll discuss how to simplify the numerator in a given fraction. The numerator is the top part of a fraction and represents the number of parts we have. For the expression \left(\frac{9 + 2 \cdot 0}{9}\right)^{2}, the numerator is initially \[ 9 + 2 \cdot 0 \]. To simplify it:

• First, perform any multiplication within the numerator. Here, \(2 \cdot 0\) equals 0 because any number multiplied by zero is zero.
• Then, add the result to any remaining terms. In this case, \(9 + 0 = 9 \).

As a result, the simplified numerator of the fraction is 9.

division in fractions

Next, let's understand how division in fractions works. It is represented by the line between the numerator and the denominator. The fraction \frac{9}{9} means that the number 9 is divided by 9. Here’s how you can think about it:

• Place the simplified numerator over the denominator. In our example, it's \frac{9}{9} \.
• Perform the division: \[ \frac{9}{9} = 1 \]. Division tells us how many times the denominator fits into the numerator. Since 9 divided by 9 equals 1, our fraction simplifies to 1.

Fractions are fundamentally a way to represent division. Always make sure to simplify fractions completely, as it often makes subsequent calculations more straightforward.

squaring numbers

Finally, we'll cover squaring numbers, which is raising a number to the power of 2. It is represented by an exponent of 2. For example, \(1^2 = 1 \times 1\). Here's a simple breakdown:

• First, identify the number you need to square. In this case, it’s 1 from our previous result of the division.
• Multiply that number by itself: \[ 1 \times 1 = 1 \].

Squaring is simply repeated multiplication of a number by itself. This operation often appears in algebraic expressions, solving equations, and understanding geometric concepts. Note that numbers greater than 1 will grow, numbers between 0 and 1 will shrink, and both 0 and 1 remain the same when squared.