Problem 26 - Algebra 1

Question

Evaluate the expression. Write fractions in simplest form. $$ \left(\frac{3}{4}\right)^{2} $$

Step-by-Step Solution

Answer

\(\frac{9}{16}\)

Step 1: Understanding the exponent

To square a number means to multiply it by itself. This also applies when squaring fractions.

Step 2: Squaring the fraction

Squaring the fraction \(\frac{3}{4}\) gives \((\frac{3}{4})^2 = \frac{3^2}{4^2}\)

Step 3: Calculating the square

Next, calculate the square of the numerator and the denominator separately. \(3^2 = 9\) and \(4^2 = 16\)

Step 4: Writing the result

The final result is \(\frac{9}{16}\)

Key Concepts

Squaring Fractions

When working with fractions, knowing how to square them is an essential skill. To square a fraction, simply multiply the fraction by itself. This is denoted by the exponent 2, which tells us how many times to use the fraction in a multiplication. It's important to note that squaring a fraction involves both the numerator (the top number) and the denominator (the bottom number).

For example, let's square \( \frac{3}{4} \). Follow these steps:

• Write the fraction twice, because squaring means multiplying the number by itself: \( \frac{3}{4} \times \frac{3}{4} \).
• Square the numerator and the denominator separately: \( 3^2 = 9 \) and \( 4^2 = 16 \).
• Write the result as a new fraction: \( \frac{9}{16} \).

This process is paramount in algebra and can be applied to any fractional value. Remember, the key is to perform the operation independently on the numerator and denominator before combining them back into a single fraction.

Exponents in Algebra

Exponents are often encountered in algebra and indicate how many times a number is multiplied by itself. They are written as small numbers to the upper-right of the base number. When dealing with fractions, exponents apply to both the numerator and denominator. In algebra, it's critical to understand how exponents affect different types of numbers, including fractions, integers, and variables.

Here's a simple guideline for applying exponents to fractions:

• The exponent outside the parentheses raises both the numerator and denominator to that power.
• If you have \( \left(\frac{a}{b}\right)^{n} \), square both 'a' and 'b' independently, leading to \( \frac{a^n}{b^n} \).
• Algebraic properties of exponents, like \( (ab)^{n} = a^{n}b^{n} \), also apply to fractions.

Grasping exponents in algebra simplifies working with more complex expressions and solving algebraic equations efficiently.

Simplest Form

Writing a fraction in its simplest form means finding an equivalent fraction where the numerator and denominator are as small as possible, and have no common factors other than 1. This often involves dividing both by their greatest common divisor (GCD).

When simplifying the fraction \( \frac{9}{16} \) from our earlier example, we observe that 9 and 16 do not have any common factors other than 1, which tells us that this fraction is already in its simplest form. For other fractions, a process of factoring may be required to find the GCD. Simplifying is important because it makes fractions easier to understand and work with, especially when adding, subtracting, or comparing them. Remember, the simplest form of a fraction helps to clearly see the ratio it represents, which can be crucial in various applications in mathematics and real-world problems.