Problem 2 - Algebra 2 and Trigonometry

Question

Natasha said that \((2)^{3}(5)^{3}=(10)^{3} .\) Do you agree with Natasha? Justify your answer.

Step-by-Step Solution

Answer

Yes, I agree with Natasha; \((2)^3 (5)^3 = (10)^3\) because both simplify to 1000.

Step 1: Understand the Given Equality

Natasha claims that \((2)^3 (5)^3 = (10)^3\). This means she believes the product of two powered terms \((2)^3\) and \((5)^3\) equals another powered term \((10)^3\).

Step 2: Calculate \((2)^3\)

Calculate \((2)^3\) which means raising 2 to the power of 3: \(2 \times 2 \times 2 = 8\).

Step 3: Calculate \((5)^3\)

Calculate \((5)^3\) which means raising 5 to the power of 3: \(5 \times 5 \times 5 = 125\).

Step 4: Calculate \((10)^3\)

Calculate \((10)^3\) which means raising 10 to the power of 3: \(10 \times 10 \times 10 = 1000\).

Step 5: Calculate \((2)^3 (5)^3\)

Multiply the results of \((2)^3\) and \((5)^3\): \(8 \times 125 = 1000\).

Step 6: Compare the Two Results

Compare the result of \((2)^3 (5)^3 = 1000\) with \((10)^3 = 1000\). Both results are 1000, which means the equality \((2)^3 (5)^3 = (10)^3\) holds true.

Key Concepts

Powers of Numbers

In mathematics, raising a number to a power means multiplying that number by itself a specific number of times. The power of a number is often expressed using an exponent. For example, in the expression \((2)^3\), the number 2 is referred to as the base, and 3 is the exponent. This tells us to multiply 2 by itself three times:

• \(2 \times 2 = 4\)
• \(4 \times 2 = 8\)

Therefore, \((2)^3 = 8\). Requiring a number to be multiplied multiple times is very powerful for simplifying expressions and solving equations, as it reduces repetitive operations into a concise form. Understanding powers of numbers is essential in algebra as it allows for handling large numbers and expressions efficiently.

Multiplication

Multiplication is a fundamental operation in mathematics that combines two numbers into one product. Consider the example from the exercise: multiplying powers of numbers \((2)^3\) and \((5)^3\). Separately, these are 8 and 125 respectively. To find the product of these two expressions, you multiply the results of the powers:

• \(8 \times 125 = 1000\)

Multiplication brings together individual quantities to form a larger quantity. This operation is critical because it reflects repeated addition and is essential in understanding relationships between quantities, especially in algebra. It helps determine the collective outcome when multiple factors are involved, such as the combination of two separate powers leading to a single result.

Equality in Algebra

Equality is a central concept in algebra that signifies that two expressions represent the same mathematical value. The exercise demonstrates this through the equation \((2)^3 (5)^3 = (10)^3\). Here's the breakdown:

• Calculate each side separately: \((2)^3 \times (5)^3 = 8 \times 125 = 1000\)
• \((10)^3 = 1000\)

Both sides simplify to 1000, hence they are equal, confirming that Natasha's statement is correct. Equality signifies balance in equations and indicates that different expressions can hold the same value. In algebra, recognizing equal expressions allows us to solve equations and verify solutions effectively. It is a key tool in problem-solving, as it ensures that both sides of an equation are balanced and equivalent.