Problem 53 - Algebra A Combined Function

Question

The vault below is in the shape of a cube. If each side is \(3 y^{4}\) feet, find its volume.

Step-by-Step Solution

Answer

The volume of the cube is \( 27 y^{12} \text{ cubic feet} \).

Step 1: Understand the Volume Formula for a Cube

The volume of a cube is found using the formula \( V = s^3 \), where \( s \) is the length of one side of the cube. In this exercise, the side of the cube is given as \( 3 y^4 \).

Step 2: Substitute the Side Length

Substitute the given side length \( s = 3 y^4 \) into the volume formula. This gives us \( V = (3 y^4)^3 \).

Step 3: Apply the Law of Exponents

Apply the power of a power rule which states \((a^m)^n = a^{m\cdot n}\). First, calculate \((3)^3 = 27\). Then, \((y^4)^3 = y^{12}\).

Step 4: Compute the Expression

Combine the results from previous calculations: \( V = 27 y^{12} \). This means that the volume of the cube is \( 27 y^{12} \text{ cubic feet} \).

Key Concepts

Cube in Geometry

A cube is a three-dimensional geometric shape with six equal square faces, twelve edges, and eight vertices. All sides of a cube are of equal length, and angles between any two adjacent faces are right angles by definition. This simplicity makes cubes a popular subject in geometry.

One of the defining characteristics of a cube is that its volume can be calculated easily. The volume is simply the length of one of its edges raised to the power of three because the formula for the volume of a cube is given by:

• The volume (\[ V \]) = edge length (\[ s \]) cubed, or \[ V = s^3 \].

Understanding this characteristic helps when solving problems like the one in our exercise where it involves finding the volume using given side lengths. This geometry serves as a foundation for many other concepts in mathematics and science.

Understanding the Law of Exponents

The law of exponents is a crucial algebraic rule that simplifies complex expressions with powers. Specifically, it tells us how to handle expressions raised to another power. The formula is:

• If you have an expression \( (a^m)^n \), the law states that you multiply the exponents: \( a^{m \cdot n} \).

In the exercise, we used the law of exponents to simplify \( (y^4)^3 \) to \( y^{12} \). This powerful rule allows us to break down and simplify even the most intricate expressions into manageable components.

Other useful laws include:

• Product of powers: \( a^m \cdot a^n = a^{m+n} \).
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a product: \( (ab)^n = a^n \cdot b^n \).

Mastery of these rules is critical in algebra and beyond for simplifying expressions, solving equations, and understanding functions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form while keeping their value unchanged. This process often requires us to use mathematical properties and operations, such as combining like terms, factoring, or applying the law of exponents.

The goal is to transform a complex expression into a more usable form. For example, in the given exercise, the expression \( (3y^4)^3 \) was simplified into \( 27y^{12} \) by calculating powers separately and then combining them. First, \( 3^3 \) yielded \( 27 \) and \( (y^4)^3 \) became \( y^{12} \).

Here are steps often involved:

• Identify like terms or expressions raised to the same power and combine them.
• Apply laws of exponents to manage powers effectively.
• Simplify coefficients (numerical constants) by performing arithmetic operations.

Simplifying expressions is not only important in finding solutions but also in helping to identify patterns, work with functions, and understand mathematical models efficiently.