Question

Write with a single exponent. $$ \left((a+b)^{2}\right)^{5} $$

Short answer

Question: Simplify the following expression: $((a+b)^2)^5$ Answer: $(a+b)^{10}$

Step-by-step solution

Identify the base and exponents

In our given expression, the base is \((a+b)\), and the two exponents are \(2\) and \(5\). So we can represent it like \((a^m)^n\) where \(a=(a+b)\), \(m=2\), \(n=5\).

Apply the exponent property

Using the property of exponents \((a^m)^n = a^{mn}\), we can multiply the exponents \(m\) and \(n\) to get the single exponent. Here, multiplying exponents \(2\) and \(5\), we get: \(2*5=10\).

Rewrite the expression with the single exponent

Now that we found the new exponent, we'll write the base \((a+b)\) raised to this new exponent, which is \(10\). So the expression can be simplified as \((a+b)^{10}\). The final simplified expression is: $$ \left((a+b)^{2}\right)^{5} = (a+b)^{10} $$

Key concepts

Properties of Exponents

Exponents are mathematical tools that help us express repeated multiplication of the same number or expression. They play a key role in algebra and are governed by specific properties. Understanding these properties can simplify complex expressions conveniently.

One fundamental property of exponents is the **power of a power rule**. This rule states that when an exponential term is raised to another exponent, you multiply the exponents. Mathematically, it is expressed as

• For any base \(a\) and exponents \(m\) and \(n\), \[(a^m)^n = a^{m \cdot n}\].

Using this property, you can transform expressions by combining their exponents into a single term.

This becomes particularly useful when dealing with nested exponents, as it simplifies calculations and expressions, making them easier to understand and work with.

Simplifying Expressions

Simplifying expressions in mathematics allows for an easier manipulation and understanding of terms. The aim is to condense an expression into its simplest form by applying algebraic rules. In the context of exponents, simplification often involves reducing multiple layers of powers to a single term.

Take for example the expression \[\left((a+b)^2\right)^5\].Instead of having multiple exponents like 2 and 5, we merge them using known properties of exponents.

• First, identify the parts of the expression: base \( (a+b) \) and exponents \(2\) and \(5\).
• Next, apply the power of a power rule to combine those exponents into one.
• Finally, rewrite the expression using the new exponent to reflect its simplified form.

Thus, the original expression is simplified to \( (a+b)^{10} \), representing the terms in an efficient and compact form.

Exponent Rules

Exponent rules are guidelines that help navigate through calculations involving powers. They can transform expressions and make operations manageable. Here are some essential rules to remember:

• **Product of Powers Rule**: To multiply terms with the same base, add their exponents: \[a^m \cdot a^n = a^{m+n}\].
• **Quotient of Powers Rule**: To divide terms with the same base, subtract their exponents: \[\frac{a^m}{a^n} = a^{m-n}\].
• **Power of a Product Rule**: A product raised to an exponent is the product of each factor raised to the exponent: \[(ab)^m = a^m \cdot b^m\].
• **Power of a Power Rule** (as applied in our solution): Multiply exponents when raising a power to another power: \[(a^m)^n = a^{m \cdot n}\].

Each of these rules allows you to manipulate and simplify expressions methodically, ensuring you can resolve complex problems step-by-step.