Problem 17 - Algebra A Combined Function

Question

Evaluate. $$ (0.7)^{3} $$

Step-by-Step Solution

Answer

(0.7)^3 equals 0.343.

Step 1: Understanding the Problem

The given problem is to evaluate \((0.7)^{3}\). This means we need to multiply 0.7 by itself three times.

Step 2: Multiply 0.7 by itself (First Multiplication)

First, calculate the product of 0.7 and 0.7. Multiply these two values: \(0.7 \times 0.7 = 0.49\).

Step 3: Multiply the Result by 0.7 (Second Multiplication)

Now, take the result from the previous step, 0.49, and multiply it by 0.7 again to complete the required cubed power operation: \(0.49 \times 0.7 = 0.343\).

Step 4: Conclusion

After performing both multiplications, we conclude that \((0.7)^{3} = 0.343\).

Key Concepts

Multiplication

Multiplication is one of the four basic operations in arithmetic. It involves finding the total number from combining several equal groups. When we multiply two numbers, we essentially add one number to itself a specific number of times indicated by the other number. For example, if we want to multiply 3 by 4, it means we are adding 3 four times: \(3 + 3 + 3 + 3 = 12\).
This is why multiplication is often described as repeated addition. It's a more efficient way to add the same number multiple times.

• In the context of our original exercise, when we multiply \(0.7\) by itself, we are essentially creating three groups of \(0.7\).
• The first step in the multiplication process involves multiplying \(0.7\) by \(0.7\) to get \(0.49\). This is the first instance of repeated addition, combining two groups of \(0.7\).
• Then, we take the result \(0.49\) and multiply it by \(0.7\) one more time to include the third group, resulting in \(0.343\).

This example shows how multiplication is pivotal in calculating powers of numbers.

Cubing Numbers

Cubing a number involves raising it to the power of three. This means we multiply the number by itself, then multiply the result by the original number again. In mathematical terms, cubing a number \(x\) is represented as \(x^3\).
In simpler terms, it's like stacking three layers of the same number together.

• Let's take \(0.7\) as an example. Cubing \(0.7\) means calculating \(0.7 \times 0.7 \times 0.7\).
• First, \(0.7 \times 0.7 = 0.49\), and then \(0.49 \times 0.7 = 0.343\), which gives us the final result of \(0.343\) for \((0.7)^{3}\).
• The concept of cubing is widely used in volume calculations, where you often cube a side length to find the volume of a cube.

Understanding how to cube numbers is a fundamental part of grasping exponent rules, as it lays the foundation for more complex operations involving powers of numbers.

Algebra Basics

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, we often use letters to represent numbers. This allows us to create formulas and equations that can solve various mathematical problems.
Even basic problems like calculating \((0.7)^{3}\) rely on algebraic principles.

• Exponentiation is a crucial algebraic concept where we multiply a number by itself repeatedly. For cubing, the exponent is \(3\).
• Algebra teaches us the properties of exponents, such as \(x^{a} \times x^{b} = x^{a+b}\). Understanding these rules helps simplify and solve algebraic expressions.
• In our example, knowing that exponentiation involves repeated multiplication is an application of these basic algebra ideas.

Mastering algebraic basics is essential for tackling more advanced topics in mathematics. It provides the language and tools needed to express mathematical ideas clearly and effectively.