Problem 77 - Elementary Algebra

Question

Simplify \(\left[\frac{2 x^{8}(x-1)^{5}}{x^{4}(x-1)^{2}}\right]^{4}\).

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left[\frac{2 x^{8}(x-1)^{5}}{x^{4}(x-1)^{2}}\right]^{4}\). Answer: \(16x^{16}(x-1)^{12}\)

Step 1: Apply the power of a quotient rule:

First, we need to apply the power of a quotient rule, which states that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). Therefore, we can rewrite the expression as: $$\left[\frac{2^4 x^{8\cdot4}(x-1)^{5\cdot4}}{x^{4\cdot4}(x-1)^{2\cdot4}}\right]$$

Step 2: Simplify the exponents

Next, we need to simplify the exponents in the expression. This gives us: $$\frac{2^4 x^{32}(x-1)^{20}}{x^{16}(x-1)^{8}}$$

Step 3: Apply the exponent laws for multiplication and division

Now, let's apply the exponent laws for multiplication and division. To do this, we'll subtract the exponents of the common bases (\(x\) and \((x-1)\)) in the numerator and the denominator. For \(x\), we have: \(32-16 = 16\) For \((x-1)\), we have: \(20-8 = 12\) So, the expression simplifies to: $$2^4 x^{16}(x-1)^{12}$$

Step 4: Final simplification

Finally, we simplify the expression by calculating \(2^4\), which is equal to \(16\). So, the final simplified expression is: $$\boxed{16x^{16}(x-1)^{12}}$$

Key Concepts

Exponent Rules

Exponent rules are fundamental in algebra, enabling us to simplify expressions involving powers effectively. One of the primary rules is the product of powers rule, which dictates that when you multiply similar bases, you add their exponents:

• \((a^m)(a^n) = a^{m+n}\).

Another essential rule is the quotient of powers, which states that dividing similar bases involves subtracting their exponents:

• \(\frac{a^m}{a^n} = a^{m-n}\).

These rules allow for the consolidation of terms and simplify complex expressions by reducing multiple powers into single terms. It's important to practice these concepts, as they become the foundation for solving problems involving higher algebra.

Power of a Quotient

The Power of a Quotient rule states that when you raise a fraction to a power, you can apply the exponent to both the numerator and the denominator separately. This is written as:

• \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).

This rule is crucial while simplifying expressions involving fractions and powers, as it enables you to separate the complexities of the numerator and the denominator. For instance, if you have \(\left(\frac{2}{3}\right)^3\), this becomes \(\frac{2^3}{3^3}\), simplifying further to \(\frac{8}{27}\).
Understanding this principle is particularly useful in algebra, where expressions often need to be simplified for ease of calculation and clarity.

Simplifying Expressions

Simplifying expressions involves combining like terms and using math principles to reduce them to their simplest form. This often includes applying exponent rules, distributing powers across factors, and eliminating any complexities in the expression. For example, in the expression \(\left[\frac{2 x^{8}\cdot(x-1)^{5}}{x^{4}\cdot(x-1)^{2}}\right]^4\), it’s crucial to distribute the power of 4 across each term to simplify it further.
Starting with simplification ensures clarity and accuracy, especially when dealing with multiple terms with variables. Each step follows logically from the previous one, using methods such as the power of a quotient and exponent rules. Simplified expressions are more manageable and reveal the underlying relationships between variables, which are beneficial for problem-solving in algebra.

Multiplication and Division of Exponents

In algebra, multiplying and dividing exponents is a common and valuable skill. When multiplying exponents with the same base, maintain the base and add the exponents:

• \((a^m)(a^n) = a^{m+n}\).

When dividing, the rule is to subtract the exponent in the denominator from the exponent in the numerator:

• \(\frac{a^m}{a^n} = a^{m-n}\).

These operations are straightforward but powerful tools for simplifying expressions. For example, given \(x^{32}/x^{16}\), using division of exponents simplifies the expression to \(x^{16}\).
Enhancing your proficiency with these rules helps in solving algebraic expressions gracefully and demonstrates the flexibility of exponents in various mathematical operations.