Question

Simplify the expression.\(\frac{x^{n} \cdot x^{2 n}}{x^{3 n}}\)

Short answer

The simplified version of the given expression \(\frac{x^{n} \cdot x^{2 n}}{x^{3 n}}\) is 1.

Step-by-step solution

Use the rule for multiplication of exponents

The expression is \(\frac{x^{n} \cdot x^{2 n}}{x^{3 n}}\). Using the rule of multiplying powers with the same base, we add the exponents, which gives us \(\frac{x^{n+2n}}{x^{3n}}\). Hence, the expression simplifies to \(\frac{x^{3n}}{x^{3n}}\).

Utilize the rule for division of exponents

In this step, we apply the rule for dividing powers with the same base. This rule says to subtract the exponents. The simplified fraction from step 1 is \(\frac{x^{3n}}{x^{3n}}\). Here, the exponents are the same, hence their subtraction will result in zero i.e., \(x^{3n-3n}\) simplifies to \(x^{0}\).

Any variable raised to the power of zero

Any variable (except zero) raised to the power of zero equals 1. Therefore, the final simplified version of the original expression is 1.

Key concepts

Exponent Rules

Understanding exponent rules is essential when dealing with powers in algebra. Exponents signify repeated multiplication. For instance, when we see an expression like \( x^3 \), it means \( x \) is multiplied by itself three times: \( x \times x \times x \). One particular rule that makes simplifying expressions more manageable is the 'Product of Powers' rule. If you multiply two powers with the same base, you can simply add the exponents. For example, \( x^a \times x^b = x^{a+b} \). This rule helps combine like terms and is especially useful in more complex equations.

Another rule, the 'Power of a Power' rule, says that when you raise a power to another power, you multiply the exponents: \( (x^a)^b = x^{a \times b} \). Understanding and applying these exponent rules correctly can greatly simplify algebraic expressions.

Division of Exponents

When you divide expressions that have exponents with the same base, you can apply the 'Quotient of Powers' rule. This rule states that you should subtract the exponent of the denominator from the exponent of the numerator: \( \frac{x^a}{x^b} = x^{a-b} \). It's crucial that the base of the exponents remains the same for this rule to be valid.

In our textbook exercise, \( \frac{x^{n} \cdot x^{2n}}{x^{3n}} \) simplifies to \( x^{3n-3n} \) when using this rule, which further simplifies the expression significantly. Being adept at applying this rule speeds up the process of simplification and helps in solving algebraic expressions efficiently.

Variables to the Power of Zero

An intriguing aspect of exponents is the concept of any nonzero variable raised to the power of zero. Regardless of what the variable or its exponent were prior, if it's raised to the power of zero, the result is always 1: \( x^0 = 1 \). This rule is universal for all nonzero numbers and variables. It often appears during simplification processes.

For example, in the process of simplifying \( \frac{x^{n} \cdot x^{2n}}{x^{3n}} \), we arrive at the expression \( x^0 \) due to the 'Quotient of Powers' rule. It's crucial to remember that \( 0^0 \) and expressions with zero in the denominator are undefined, so this rule does not apply in those cases. Otherwise, it's a simple yet powerful tool that can make complex algebraic expressions much more manageable.