Problem 19 - Algebra 1: Concepts and Skills

Question

A square painting measures 5 feet by 5 feet. Write the power that gives the area of the painting. Then evaluate the power.

Step-by-Step Solution

Answer

The power that gives the area of the painting is \(5^2\) and when evaluated, it equals to 25 square feet.

Step 1: Identifying the side of the square

The problem states that it is a square painting measuring 5 feet by 5 feet, meaning each side of this square is 5 feet.

Step 2: Writing the power to determine the area

Since the formula for the area of a square is \(side^2\), you substitute 5 (the side length) into this formula, which leads to \(5^2\)

Step 3: Evaluating the power

After substituting the value into the formula, solve the power \(5^2\). The result will be 25.

Key Concepts

Powers and Exponents

Understanding powers and exponents is essential when working with mathematical expressions. An exponent indicates how many times a number, known as the base, is multiplied by itself. This operation is a shortcut to repeated multiplication. For example, in the expression \(5^2\), 5 is the base and the exponent is 2, meaning 5 is multiplied by itself once, resulting in \(5 \times 5\).

There are a few basic rules associated with powers and exponents:

• Multiplying Powers: When two powers have the same base, you add the exponents, such as \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When there is an exponentiation of another exponent, you multiply the exponents, like \((a^m)^n = a^{m \times n}\).
• Zero Exponent: Any base raised to the exponent zero equals 1, e.g., \(a^0 = 1,\) provided \(a eq 0\).

These rules help solve more complex expressions involving exponents and are critical for evaluating the power in formulas like the area of a square.

Evaluating Powers

Evaluating powers means calculating the result of a number raised to an exponent. It involves multiplying the base by itself as many times as indicated by the exponent. This is especially useful in various mathematical formulas, like the formula for the area of a square.

To evaluate \(5^2\), for instance, start by identifying the base (5), and the exponent (2). Multiply the base by itself: \(5 \times 5 = 25\). Thus, \(5^2 = 25\). You can clearly see this simplifies repeated multiplication into a single step calculation.

If you're evaluating higher powers like \(5^3\), you'd simply multiply the base three times: \(5 \times 5 \times 5 = 125\). Evaluating powers helps streamline computations in geometry, physics, and many other disciplines.

Geometry Formulas

Geometry formulas are mathematical expressions used to calculate properties of geometric figures, such as area, volume, or perimeter. One of the most fundamental geometry formulas is for calculating the area of a square, denoted as \(side^2\). This formula represents the space inside the square by multiplying its side length by itself.

For polygons like squares and rectangles, the area formula is straightforward due to their regular shapes. For a square, each side is the same length, simplifying the calculation. Suppose a square has a side length of 5 feet, using the formula \(side^2\), the area is \(5^2 = 25\) square feet.

Geometry formulas extend beyond just area. They include other calculations, such as:

• Perimeter of a Square: \(4 \times side,\) which is the total length around the square.
• Volume and Surface Area of Solids: Complex shapes like cubes and spheres have their own formulas, such as \(side^3\) for a cube's volume.

These formulas are vital for practical applications in architecture, design, and everyday problem-solving.