Problem 54 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(\left(5 a^{4}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified form of \(\left(5 a^{4}\right)^{2}\) is \(25a^{8}\).

Step 1: Identify the Base and Exponents

In the expression \(\left(5 a^{4}\right)^{2}\), the base under both exponents is \(a\), the first exponent is 4 (which is for the base \(a\)), and the second exponent is 2 (which is for the entire expression inside the parentheses i.e \(5 a^{4}\)). The coefficient of \(a^{4}\) inside the parentheses is 5.

Step 2: Apply the Power of a Power Rule

According to the power of a power rule, when an exponent is raised to another exponent, we multiply the exponents. In the expression \(\left(5 a^{4}\right)^{2}\), the base \(a\) is raised to the exponent 4, and this result is further raised to the exponent 2. Therefore, we multiply the exponents 4 and 2 to obtain an exponent of 8 for the base \(a\). The coefficient is also squared, 5*5=25. So, \( \left(5 a^{4}\right)^{2} = 25a^{8} \).

Key Concepts

Power of a Power Rule

Exponentiation can sometimes seem confusing, but the "power of a power rule" is a straightforward principle that simplifies expressions where exponents are stacked.
When you have an expression like \((a^m)^n\), the power of a power rule tells us that you need to multiply the exponents: \(a^{m \cdot n}\).
This means for every unit of the outer exponent \(n\), you apply the inner exponent \(m\) to your base \(a\).

• In our original example, \(\left(5a^4\right)^2\), notice that the exponent 4 is applied to the base \(a\) first.
• This expression is then raised again by an exponent of 2, signifying the entire term must be squared, following our power rule.
• This converts the equation to \(a^{4 \times 2} = a^8\), demonstrating the power of this rule in handling complex algebraic expressions.

The coefficient, in this case 5, must also be considered separately when applying the rule, as it also needs to be raised to the power indicated. So, \(5^2 = 25\).
Hence, the simplified expression becomes \(25a^8\).

Simplifying Expressions

Simplifying expressions makes them easier to understand and work with in algebra.
To simplify means to reduce an expression to its simplest, most concise form, without changing its value.
Here are a few general steps to follow:

• Identify all components, like coefficients and variables with their exponents.
• Apply relevant algebraic rules, such as the power of a power rule, to combine and reduce exponents.
• Compute any numerical operations, such as squaring coefficients or combining like terms.

After identifying the pieces of the expression, return them in a streamlined format where multiplication of powers and coefficients is clearly calculated.
For our example, \(\left(5a^4\right)^2\), we computed \(5^2\) to yield \(25\) and then used the rule to find the new power as \(a^8\), effectively simplifying the expression to \(25a^8\).
A simplified expression is often easier to use in further computations, whether they relate to solving equations or substitutions.

Algebraic Expressions

Algebraic expressions can seem daunting at first, but they are the language of algebra. These expressions are composed of numbers, variables, and arithmetic operations: addition, subtraction, multiplication, and division. For a clear understanding:

• Each part of the expression, like a number or variable, is called a "term." In \(5a^4\), 5 is the numerical coefficient, and \(a^4\) is a term known as a power of a base.
• Algebraic expressions often include exponents, which indicate how many times to multiply the base by itself.
• Simplifying these expressions often involves combining like terms and applying rules like the power of a power.

It's essential to learn how to manipulate algebraic expressions because they form the foundation of problem-solving in algebra.
In the exercise given, the expression \(\left(5a^4\right)^2\) is a simple form representing how quantities change when influenced by exponents.
Mastering operations with algebraic expressions will benefit you in more advanced topics, enhancing your mathematical fluency.This understanding helps in making higher-level math problems more manageable and less intimidating.