Problem 7 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(\left(x^{5}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \( x^{10} \).

Step 1: Identify the Base and the Exponent

The given expression is \( (x^5)^2 \). Here, the base is \( x^5 \) and the outer exponent is \( 2 \).

Step 2: Apply the Power of a Power Property

Use the property \( (a^m)^n = a^{m imes n} \). Apply this to the expression: \( (x^5)^2 = x^{5 imes 2} \).

Step 3: Simplify the Exponent

Calculate the multiplication \( 5 \times 2 \) to simplify the exponent: \( x^{5 imes 2} = x^{10} \).

Step 4: Final Simplified Expression

The simplified form of the given expression is \( x^{10} \).

Key Concepts

Power of a Power Property

When dealing with exponents, understanding the power of a power property is vital. This property allows us to simplify expressions where an exponent is raised to another exponent. In algebra, this is expressed as \((a^m)^n = a^{m \times n}\). Here’s a simple way to think about it:

• The base \(a\) remains the same throughout the operation.
• The exponents \(m\) and \(n\) are multiplied together to form a single exponent.

For instance, in the expression \((x^5)^2\), the base is \(x\), and it’s first raised to the power of 5, and then that whole expression is raised to the power of 2. Applying the power of a power property means multiplying these two exponents: \(5 \times 2\), which simplifies to give \(x^{10}\). This rule helps to greatly simplify the expression, making further algebraic manipulation easier.

Simplifying Exponents

Simplifying exponents involves reducing expressions to their simplest form. This is important when dealing with algebraic expressions, as it makes them easier to understand and solve.

• Identify the base and the exponents involved.
• Apply exponent rules, like the power of a power property.
• Combine and reduce the exponents where possible.

Let’s apply this to \((x^5)^2\): We first identify our base \(x\) and recognize it is raised to the power 5, with the entire expression raised again to the power of 2. By multiplying the exponents as explained earlier, we get \(x^{10}\). This is the simplified form, making it far more manageable in larger algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operation signs. Unlike numbers alone, expressions can contain variables which stand in for unknowns or quantities that can change.When dealing with algebraic expressions that include exponents, like \((x^5)^2\), remember:

• Variables like \(x\) can have exponents which dictate the degree or power of the variable.
• Use properties of exponents, such as the power of a power property, to simplify expressions.
• Keep each step clear by expressing one transformation at a time.

Algebra is all about finding relationships and patterns, and understanding how to simplify parts of an expression such as exponents is key. It’s a fundamental skill that aids in solving equations and understanding complex mathematical problems.