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Chapter 1: Problem 56

Simplify. \(\frac{6+9 \div 3}{3^{2}}\)

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

Evaluate Division within the Numerator

First, identify any division operations within the numerator. The expression inside is \(6 + 9 \div 3\). Solve \(9 \div 3\) first, which equals 3.
02

Simplify the Numerator

Now replace \(9 \div 3\) with 3 in the numerator: \(6 + 3 = 9\). The simplified numerator is 9.
03

Evaluate the Denominator

Calculate the value of the denominator \(3^2\). The expression \(3^2\) equals 9 because 3 multiplied by itself is 9.
04

Simplify the Fraction

Now substitute back the simplified values into the fraction: \(\frac{9}{9}\). The expression \(\frac{9}{9}\) simplifies to 1.

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Key Concepts

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

Numerator Simplification
To simplify the numerator of a fraction, we first need to tackle any operations inside it according to the order of operations. In this example, the original numerator is the expression \(6 + \frac{9}{3}\). The order of operations dictates that division comes before addition.
  • First, we perform the division \(9 \div 3\), which results in 3.
  • Next, we replace the divided portion with our result, changing \(6 + \frac{9}{3}\) to \(6 + 3\).
  • Finally, we handle the addition, \(6 + 3\), which equals 9.
Thus, the simplified numerator is 9. Remembering the order of operations helps ensure accuracy by solving operations in the correct sequence.
Denominator Calculation
The denominator of a fraction is vital to its overall value, so calculating it accurately is essential. Here the denominator is expressed as \(3^2\), which involves exponents.
  • An exponent indicates how many times a number, known as the base, is multiplied by itself. So \(3^2\) means \(3 \times 3\).
  • When we perform this multiplication, we get 9.
Therefore, the denominator calculation reveals that \(3^2 = 9\). Understanding the function of exponents can simplify your work with fractions and improve your math skills.
Fraction Simplification
Simplifying a fraction means reducing it to its simplest form while keeping the same value. Our original fraction is \(\frac{9}{9}\), and both the numerator and the denominator are equal here.
  • When both the numerator and the denominator of a fraction are identical, the fraction simplifies to 1, since dividing any number by itself equals 1.
  • This simplification step is straightforward when the numbers involved are the same, simplifying the fraction by recognizing this pattern across any similar fractions.
So, \(\frac{9}{9} = 1\). Knowing this basic rule can make simplifying fractions easier and faster. Try practicing with different numbers to become comfortable with this simplifying process.

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