Question

Write each expression without parentheses. Assume all variables are positive. $$ (c f)^{9} $$

Short answer

Question: Simplify the expression (cf)^9. Answer: c^9f^9

Step-by-step solution

Apply the product of powers property

We can simplify the expression by using the product of powers property, \((a b)^m = a^m b^m\). Applying this property to the given expression: $$ (c f)^9 = (c^9) (f^9) $$ Now we have eliminated the parentheses and simplified the expression to: $$ c^9f^9 $$

Key concepts

Product of Powers Property

The Product of Powers Property is a fundamental rule of exponentiation. It helps us to break down and simplify expressions with exponents when they involve a product. In essence, when you have an expression like \((ab)^m\), this property tells us that we can distribute the exponent \(m\) to both elements inside the parentheses. Therefore, \((ab)^m = a^m b^m\).

Imagine you have items grouped together, such as \((c f)^9\). By using the Product of Powers Property, you can simplify this by distributing the exponent 9 to both \(c\) and \(f\), resulting in \(c^9 f^9\). This technique is very useful in algebra for simplifying expressions and solving equations.

Simplifying Expressions

Simplifying expressions often involves reducing complexity by eliminating unnecessary components, such as parentheses. By applying properties from exponentiation, like the Product of Powers Property, we can transform expressions into their simplest form.

In our exercise, the expression \((c f)^9\) initially looks a bit complex with the parentheses. By applying the Product of Powers Property, the expression is simplified to \(c^9 f^9\). This transformation means that every variable is expressed clearly with its exponent, making it easier to work with. Simplification is crucial in algebra as it prepares expressions for further calculations or evaluations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that contain variables, numbers, and operation symbols. They represent real-world quantities in a flexible form, allowing us to manipulate and solve problems effectively.

Consider the expression \((c f)^9\). Here, \(c\) and \(f\) represent variables, which can stand for any number value. The exponent, 9, tells us how many times the combined terms \(c\) and \(f\) are multiplied together. When handling algebraic expressions, knowing how to apply properties like the Product of Powers helps manage these expressions efficiently. It aids in elucidating the structure and relationships within the expression, making algebra a powerful tool for solving a vast array of problems.