https://file.aiquickdraw.com/mathbot/file/1730013857299dditm7f2.jpg

The given expression is:

\[
\frac{47^2 - 3^2}{27^2 - 2 \cdot 27 \cdot 13 + 13^2}
\]

Let's simplify each part of the fraction.

### Step 1: Simplify the Numerator
The numerator is \( 47^2 - 3^2 \), which is a difference of squares. We can use the formula:
\[
a^2 - b^2 = (a - b)(a + b)
\]
Here, \(a = 47\) and \(b = 3\), so:

\[
47^2 - 3^2 = (47 - 3)(47 + 3) = 44 \cdot 50 = 2200
\]

### Step 2: Simplify the Denominator
The denominator is:
\[
27^2 - 2 \cdot 27 \cdot 13 + 13^2
\]
This expression is in the form of \( a^2 - 2ab + b^2 \), which can be factored as:
\[
(a - b)^2
\]
where \( a = 27 \) and \( b = 13 \). Thus:

\[
27^2 - 2 \cdot 27 \cdot 13 + 13^2 = (27 - 13)^2 = 14^2 = 196
\]

### Step 3: Simplify the Fraction
Now we substitute the values we found:

\[
\frac{47^2 - 3^2}{27^2 - 2 \cdot 27 \cdot 13 + 13^2} = \frac{2200}{196}
\]

Dividing both the numerator and the denominator by their greatest common divisor, 4:

\[
\frac{2200}{196} = \frac{550}{49}
\]

### Final Answer
\[
\frac{47^2 - 3^2}{27^2 - 2 \cdot 27 \cdot 13 + 13^2} = \frac{550}{49}
\]

Would you like further details or have any questions?

Here are some related questions:
1. What is the difference of squares and how does it simplify expressions?
2. How can we recognize and factor expressions as perfect squares?
3. What are other methods to simplify large square terms?
4. How do you find the greatest common divisor for simplification?
5. What are other factoring techniques useful for simplifying expressions?

**Tip:** Recognizing patterns like the difference of squares or perfect square trinomials can simplify complex algebraic expressions quickly.

Simplify \( \frac{47^2 - 3^2}{27^2 - 2 \cdot 27 \cdot 13 + 13^2} \).

Learn to simplify the algebraic expression \( \frac{47^2 - 3^2}{27^2 - 2 \cdot 27 \cdot 13 + 13^2} \) by factoring and applying the difference of squares and perfect square trinomial concepts. Suitable for Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation