Problem 34 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ x^{3} \cdot \frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is x

Step 1 - Identifying the properties of exponents

Recognize that this problem requires using the quotient rule to simplify an expression with exponents. The quotient rule says that for any non-zero number a and integers m and n, \(a^{m} / a^{n} = a^{m-n}\). In this exercise, \(x^{3} / x^{2}\) translates to \(x^{3-2}\).

Step 2 - Applying the quotient rule

Once the quotient rule has been identified and understood, we can apply it to the expression \(x^{3} / x^{2}\). Using the quotient rule, we can subtract the exponent of the denominator (2) from the exponent of the numerator (3). Thus, the expression becomes \(x^{3-2}\).

Step 3 - Simplifying the expression

After applying the quotient rule, simplify the expression. \(x^{3-2}\) becomes \(x^{1}\), which can be written simply as x.

Key Concepts

Quotient Rule of Exponents

When dividing expressions with the same base, we often encounter exponents. To simplify such expressions, we use the quotient rule of exponents. This rule states that when you divide two exponents with the same base, you can subtract the exponent of the denominator from the exponent of the numerator.

For instance, given a problem like \(x^{3} \cdot \frac{1}{x^{2}}\), you would subtract the exponents: \(3 - 2 = 1\), resulting in \(x^{1}\), or simply \(x\). The base \(x\) remains unchanged. It’s crucial not just to understand this rule, but to also directly apply it to the expression to simplify it correctly.

This rule is incredibly helpful as it allows us to handle complex expressions efficiently. Always remember that the quotient rule only applies when the bases in the numerator and the denominator are identical.

Properties of Exponents

Exponents aren’t just random numbers perched above others; they embody a set of algebraic rules known as the properties of exponents. These properties govern how to manipulate expressions with exponents, and understanding them is key to simplifying algebraic expressions effectively.

Some essential properties include:

• The Product Rule (\(a^{m} \cdot a^{n} = a^{m+n}\)) – When multiplying same base numbers, add the exponents.
• The Power Rule (\((a^{m})^{n} = a^{mn}\)) – When raising a power to another power, multiply the exponents.
• The Zero Exponent Rule (\(a^{0} = 1\)) – Any number (except zero) to the power of zero equals one.

These properties serve as tools to simplify expressions, making them easier to work with. For example, when faced with \(x^{3} \cdot \frac{1}{x^{2}}\), we leverage the quotient rule, which is derived from the properties of exponents, to arrive at a simpler expression.

Negative Exponents

We encounter negative exponents in algebra and they can initially seem confusing. However, the concept is straightforward: a negative exponent simply means taking the reciprocal of the base raised to the positive of that exponent.

For example, \(x^{-n} = \frac{1}{x^{n}}\). If you are simplifying an expression with a negative exponent, you'd typically rewrite it with a positive exponent to avoid leaving any negative exponents in your final answer.

In our exercise, there was no negative exponent to begin with, but if there were, you would apply the rule by flipping the base to the opposite side of the fraction line and then simplifying further. This ties back into the properties of exponents and makes expressions cleaner and numerically unequivocal.