Problem 1 - Elementary Algebra

Question

Simplify each expression using the power rule for powers. $$ \left(x^{5}\right)^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression using the power rule: \((x^5)^4\) Answer: The simplified expression is \(x^{20}\).

Step 1: Apply the Power Rule for Powers

We will apply the power rule for powers, \((a^m)^n = a^{mn}\), to the given expression \((x^5)^4\). Here, a = x, m = 5, and n = 4.

Step 2: Calculate the Simplified Exponent

Using the power rule, we calculate the simplified exponent: mn = 5*4 = 20.

Step 3: Write the Simplified Expression

Now that we have the simplified exponent, we can rewrite the expression with the simplified exponent: \(x^{20}\). The simplified expression is: $$ x^{20} $$

Key Concepts

Power Rule

In algebra, the power rule is a fundamental concept that helps us manage expressions involving exponents. Specifically, when we encounter an expression like

• \((a^m)^n\)

we apply the power rule to simplify it. The power rule states that:

• \((a^m)^n = a^{m\cdot n}\)

This means that you multiply the exponents together. It's like saying that if you have a power raised to another power, you can simplify it by multiplying those powers. This rule simplifies complex exponent expressions and makes calculations easier by breaking a complicated expression into simpler components.
For example, in the expression \((x^5)^4\), applying the power rule helps us quickly find that the simplified form is \(x^{20}\). This makes it clear that the power rule is an efficient way to handle nested exponents.

Exponents

Exponents are a mathematical way of expressing repeated multiplication of the same number. When you see a number like \(x^5\), it indicates that \(x\) is multiplied by itself 5 times.
The number \(x\) is the base, and the number 5 is the exponent. Exponents play a crucial role in simplifying expressions and solving algebraic problems.

• The base is the main number being multiplied.
• The exponent tells you how many times to multiply the base by itself.

Exponents follow specific rules, like the power rule, which we use to simplify expressions efficiently. Understanding exponents helps in addressing problems across various mathematical areas, from algebra to calculus.
In the simplification process, like in the example \((x^5)^4\), knowing how to manipulate exponents using the power rule allows us to go from a complicated expression to a simpler one, such as \(x^{20}\), in just a few steps.

Simplifying Expressions

Simplifying expressions in algebra is about making them easier to work with. This involves combining like terms, applying algebraic rules, and rewriting expressions in their most concise form. With exponents, simplifying often involves using rules like the power rule for easy manipulation.
For instance, simplifying \((x^5)^4\) involves using the power rule to multiply the exponents, resulting in the expression \(x^{20}\).

• Identify the rules applicable to the parts of an expression.
• Rearrange and combine terms accordingly.
• Simplify step by step to ensure clarity and avoid mistakes.

Simplifying allows for easier calculation, understanding, and application of the expression in larger equations or problems.
Moreover, it helps students build confidence in solving algebraic problems, as it turns complex tasks into manageable ones by breaking them down into smaller, understandable parts.