Problem 65 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Write your answer as a power. $$ (7 x)^{2} $$

Step-by-Step Solution

Answer

The simplified expression for \((7x)^2\) is \(49x^2\).

Step 1: Apply the rule of exponentiation to the problem

According to the rule of exponents, the power can be distributed to each part of the product separately. This means that \((7x)^2\) is equal to \(7^2 * x^2\).

Step 2: Solve the Powers

Now calculate the powers separately. \(7^2 = 49\) and \(x^2\) remains as it is since we can't calculate the power of a variable.

Step 3: Write the simplified expression

Now, combine the results from Step 2 to rewrite the simplified expression. This gives you \(49x^2\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers, called base and exponent. The base is the number being multiplied by itself, while the exponent indicates how many times it is being multiplied. For instance, in \(7^2\), 7 is the base, and 2 is the exponent, meaning 7 is multiplied by itself 2 times.

• \(a^m \) means 'a' is multiplied by itself 'm' times.
• For numbers, we can directly calculate the result.
• For variables, such as \(x^2\), we express the exponentiation as a power since exact calculation isn't possible without further context.

Exponents make it easier to write and compute large products concisely. In expressions like \((7x)^2\), the exponentiation applies to both the number and the variable involved.

Distributive Property of Exponents

The distributive property of exponents is useful when dealing with expressions raised to a power. It tells us how exponents apply across multiplication inside brackets. When you have a term like \((ab)^n\), you can distribute the exponent to each component of the product:

• \((ab)^n = a^n \cdot b^n\)
• Each part of the multiplication inside the parentheses gets its own exponent.

This property simplifies the expression by breaking it into separate components. In the example \((7x)^2\), the rule allows us to rewrite it as \(7^2 \cdot x^2\). So each part of the expression, both the constant and the variable, is treated individually.

Simplifying Expressions

Simplifying expressions is the process of making them easier to work with. By applying the rules of arithmetic and properties of exponents, you can reduce expressions to their simplest form. Take for example the expression \((7x)^2\):

• Use the distributive property of exponents to expand it into \(7^2 \cdot x^2\).
• Calculate \(7^2\) to get 49, keeping \(x^2\) as is because x represents an unknown value.
• Combine the results to arrive at the simplified form, \(49x^2\).

Simplifying expressions ensures that calculations are straightforward and results are presented in a compact form. This step-by-step approach brings clarity, especially when simplifying more complex algebraic expressions.