Problem 72 - 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. $$ x^{3}\left(\frac{x^{6}}{x^{2}}\right) $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(x^7\).

Step 1: Apply the quotient rule of exponents

Recall that the quotient rule of exponents states that \(\frac{a^{m}}{a^{n}} = a^{m-n}\). Using this rule, we can simplify the expression inside the parentheses: $$\frac{x^{6}}{x^{2}} = x^{6-2} = x^4$$ Now our expression is: $$x^{3}(x^4)$$

Step 2: Apply the product rule of exponents

The product rule of exponents states that \(a^{m}*a^{n} = a^{m+n}\). Applying this rule, we get: $$x^{3}(x^4) = x^{3+4} = x^7$$ So, our simplified expression is: $$x^7$$

Key Concepts

Product Rule of Exponents

When multiplying two exponential expressions that have the same base, the product rule of exponents allows us to add the exponents together. Mathematically, this is expressed as:
\[ a^m \times a^n = a^{m+n} \]
This rule makes it easier to work with powers of the same base by combining them into one single term. For instance, when simplifying \( x^3 \times x^4 \), we simply add the exponents 3 and 4 to get \( x^{3+4} = x^7 \). It’s crucial to remember that the bases must match for this rule to apply.

Using this rule doesn't just shorten the expression; it also makes computations more manageable and prepares the groundwork for further algebraic operations.

Quotient Rule of Exponents

Dividing exponential expressions with the same base calls for the quotient rule of exponents. This rule instructs us to subtract the exponent of the denominator from the exponent of the numerator:
\[ \frac{a^m}{a^n} = a^{m-n} \]
For example, when simplifying \( \frac{x^6}{x^2} \), we subtract 2 from 6, which simplifies to \( x^{6-2} = x^4 \). The assumption here is that the base \(a\) is not zero since division by zero is undefined in mathematics. Also, it’s important to ensure the exponents are whole numbers to avoid complications arising from negative or fractional exponents.

Exponent Properties

Zero and Negative Exponents

Exponents aren't just positive whole numbers. A base raised to the power of zero is always one (\(a^0 = 1\)), and a negative exponent signifies the reciprocal (\(a^{-n} = \frac{1}{a^n}\)).

Power of a Power Rule

Another key exponent property is the power of a power rule, where \((a^m)^n = a^{m \times n}\). This property is particularly useful when dealing with expressions where an exponent is already raised to another exponent.

Understanding these exponent properties helps in identifying the most efficient way to simplify an expression, whether it's by reducing powers or transforming negatives and zeros into a simpler form.

Algebraic Simplification

Simplifying algebraic expressions involves reducing them to their simplest form using a variety of techniques, including exponent rules. Algebraic simplification often means distilling expressions down to the least complicated terms possible.

For example, combining like terms, using distribution, and applying the aforementioned exponent rules are all parts of algebraic simplification. Correct and strategic use of these techniques doesn't just change the appearance of an expression – it can make the difference in solving a problem quickly and accurately.

Applying simplification rules in a considered sequence—like simplifying exponents before multiplying terms together—is an important strategy that can save time and reduce the risk of errors in more complex algebraic problems.