Problem 57 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{x^{4}}{x^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(\frac{x^{4}}{x^{2}}\) Answer: \(x^{2}\)

Step 1: Identify the base and exponents

In the given expression, the base is 'x' and the exponents are 4 and 2. The expression can be written as: $$ \frac{x^{4}}{x^{2}} $$

Step 2: Apply the quotient rule of exponents

The quotient rule of exponents states that for any nonzero base 'a' and whole number exponents 'm' and 'n', the expression \(a^{m}/a^{n}\) can be simplified as follows: $$ a^{m}/a^{ n}=a^{m-n} $$ Applying the quotient rule to the given expression, we get: $$ \frac{x^{4}}{x^{2}}=x^{4-2} $$

Step 3: Simplify the exponent

Now, subtract the exponents to get the final simplified expression: $$ x^{4 - 2}=x^{2} $$ The simplified expression is: $$ x^{2} $$

Key Concepts

Quotient Rule of Exponents

The quotient rule of exponents is a helpful rule used when you're dividing two expressions with the same base. When you see a fraction like \(\frac{a^m}{a^n}\), where both the numerator and the denominator share the same base, the rule allows you to simplify by subtracting the exponents. For example, if you have \(x^4\) divided by \(x^2\), you subtract 2 from 4, which means:

• \(a^m / a^n = a^{m-n}\)
• \(x^4 / x^2 = x^{4-2}\)

The result is \(x^2\). This rule simplifies expressions by reducing the power of the base without changing the base itself.

Product Rule of Exponents

The product rule of exponents comes into play when you multiply two powers with the same base. If you have \(a^m\) times \(a^n\), you simply add the exponents. This trick keeps the base steady while making the computation easier:

• \(a^m \cdot a^n = a^{m+n}\)
• For example, \(x^3 \cdot x^2 = x^{3+2} = x^5\).

By adding the exponents, you find that the expression simplifies in a straightforward way. This rule highlights the harmony of exponential calculations with a common base.

Simplifying Expressions

Simplifying expressions requires applying the rules of exponents, such as the quotient or product rule, to write the expression in its simplest form. Simplification means reducing an expression without losing its value, making it easier to interpret or solve. For example, using the quotient rule with \(\frac{x^4}{x^2}\) reduces it to \(x^2\).

• Look at all bases and their paired exponents.
• Apply appropriate exponent rules.
• Combine like terms if possible.

By consistently using these steps, expressions become more manageable and understandable, leading to clearer calculations and solutions.

Whole Number Exponents

Whole number exponents are exponents that are non-negative integers like 0, 1, 2, 3, and so on. They make calculations more predictable while denoting repeated multiplication of a base:

• \(x^0 = 1\) for any \(x eq 0\).
• \(x^1 = x\) means just the base itself.
• \(x^n = x \cdot x \ldots \text{(n times)}\)

Working with whole number exponents simplifies expressions by turning multiplication into a concise form. They suit everyday math problems, providing a reliable way to express repeated factors in a tidy package.