Problem 30 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{\left(x^{2}\right)^{8} x}{x^{9}} $$

Step-by-Step Solution

Answer

The simplified expression is \( x^8 \).

Step 1: Apply the Power of a Power Rule

The expression \( (x^2)^8 \) involves a power raised to another power. By the power of a power rule, \( (a^m)^n = a^{m imes n} \), so we have \( (x^2)^8 = x^{16} \). Thus, the expression becomes \( \frac{x^{16} x}{x^9} \).

Step 2: Combine Like Terms in the Numerator

Combine the terms in the numerator. Using the product of powers property, \( a^m \times a^n = a^{m+n} \). Therefore, \( x^{16} \times x = x^{17} \). The expression now is \( \frac{x^{17}}{x^9} \).

Step 3: Apply the Quotient of Powers Rule

For the expression \( \frac{x^a}{x^b} \), use the quotient of powers rule, which states \( \frac{x^a}{x^b} = x^{a-b} \). Thus, \( \frac{x^{17}}{x^9} = x^{17-9} = x^8 \).

Step 4: Write the Result with Positive Exponents

Ensure the final expression has only positive exponents. Since \( x^8 \) already uses positive exponents, no further simplification is needed.

Key Concepts

Understanding Exponents

Exponents are a crucial aspect of mathematics. They indicate the number of times a base number is multiplied by itself. For example, in the term \( x^3 \), \( x \) is the base, and \( 3 \) is the exponent. This translates into \( x \times x \times x \). Exponents simplify repeated multiplication, making it easier to handle large computations.
Exponents follow specific rules that help us simplify expressions involving multiple powers. For instance, when you see \( x^a \times x^b \), you can use the rule of adding the exponents to simplify it to \( x^{a+b} \). This is essential for working efficiently with algebraic expressions.

Exploring the Power of a Power Rule

The power of a power rule helps in simplifying expressions where we have a power raised to another power. For example, \( (x^m)^n \) simplifies to \( x^{m \times n} \). This means you multiply the exponents. For the original problem, \( (x^2)^8 \) was computed as \( x^{2 \times 8} = x^{16} \).
This rule is particularly useful when dealing with expressions in parenthesis where multiple layers of exponents are involved. It turns a potentially complex looking expression into a simpler form by reducing the number of layers you're dealing with.

Product of Powers Property

The product of powers property tells us how to handle situations where the same base appears in a multiplication. When multiplying like bases, such as \( a^m \times a^n \), we add the exponents to get \( a^{m+n} \).
In our exercise, after applying the power of a power rule, we had to simplify \( x^{16} \times x \), combining them into \( x^{17} \). Recognizing this pattern helps in executing efficient simplification, making algebraic manipulations more straightforward.

Applying the Quotient of Powers Rule

When dividing powers with the same base, you utilize the quotient of powers rule. This rule states that \( \frac{x^a}{x^b} = x^{a-b} \). Essentially, you subtract the exponent in the denominator from the exponent in the numerator.
In the given solution, after simplifying the numerator to \( x^{17} \), and having \( x^9 \) in the denominator, the rule was used to arrive at \( x^{17-9} = x^8 \). Applying this rule simplifies the division of powers significantly, resulting in a more manageable expression.