Problem 41 - Elementary Algebra

Question

\(7 \times 10^{3}+2 \times 10^{3}\)

Step-by-Step Solution

Answer

The expression simplifies to \(9 \times 10^{3}\).

Step 1: Identify the terms in the expression

The given expression is composed of two terms: \(7 \times 10^{3}\) and \(2 \times 10^{3}\).

Step 2: Factor out the common term

Both terms share a common factor of \(10^{3}\). So, you can factor out \(10^{3}\) from both terms: \(7 \times 10^{3} + 2 \times 10^{3} = (7 + 2) \times 10^{3}\).

Step 3: Add the coefficients

Add the coefficients 7 and 2: \(7 + 2 = 9\).

Step 4: Multiply the combined coefficient by the factor

Now, multiply the result by the common factor: \(9 \times 10^{3}\).

Step 5: Write the final answer

The expression simplifies to \(9 \times 10^{3}\).

Key Concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication of the same number. For example, the expression \(10^3\) means we multiply 10 by itself three times: \(10 \times 10 \times 10 = 1000\). In our exercise, both terms have the exponent \(10^3\). When you see this in an expression, it helps to recognize that these terms can be combined by using the properties of exponents.

Exponents follow specific rules:

Multiplying same bases: \(a^m \times a^n = a^{m+n}\)

Dividing same bases: \(a^m \ a^n = a^{m-n}\)

Power of a power: \((a^m)^n = a^{mn}\)

These rules are valuable tools for handling problems involving exponents, allowing you to simplify complex expressions more efficiently.

Factoring in Algebra

Factoring is a powerful algebraic technique that involves breaking down expressions into simpler components. In essence, it allows us to reverse the process of multiplication by identifying and extracting common factors.

Consider the expression \(7 \times 10^3 + 2 \times 10^3\). Here, the common factor is \(10^3\). By factoring this out, we transform the expression into \((7 + 2) \times 10^3\).

Why factor? Here are a few reasons:

Makes expressions simpler to manage

Useful for solving equations

Reveals hidden structures within expressions

In our specific case, once the common factor \(10^3\) is factored out, it simplifies the addition of the coefficients 7 and 2, making the problem much easier to solve.

Simplifying Expressions

Simplifying expressions means making them easier to work with by applying algebraic rules and operations. This step is crucial for solving algebra problems effectively.

In our exercise, after factoring out the common term \(10^3\), we are left with a simpler expression: \((7 + 2) \times 10^3\). Simplifying involves:

• Combining like terms
• Reducing fractions
• Factoring where possible

For our problem, adding the coefficients 7 and 2 gives us 9, which boils down the expression to \(9 \times 10^3\). This is a clear example of how simplifying an expression makes it more straightforward to understand and solve.

It's a strategy that helps reduce confusion and allows you to see the underlying structure of algebraic expressions vividly.