Question

Write the expression as an equivalent expression in the form \(x^{n}\) and give the value for \(n\). $$ \frac{\left(x^{3}\right)^{2}}{\left(x^{2}\right)^{3}} $$

Short answer

Question: Simplify the given expression \(\frac{(x^3)^2}{(x^2)^3}\) into the form \(x^n\) and find the value of \(n\). Answer: The simplified expression is \(x^0\), and the value of \(n\) is \(0\).

Step-by-step solution

Use the property of exponents to simplify the numerator

Recall that \((a^m)^n = a^{mn}\). Apply this rule to the numerator to simplify it: $$ \left(x^{3}\right)^{2} = x^{3 \cdot 2} = x^{6} $$

Use the property of exponents to simplify the denominator

Similarly, we apply the same rule to the denominator: $$ \left(x^{2}\right)^{3} = x^{2 \cdot 3} = x^{6} $$

Simplify the ratio using the quotient rule for exponents

Recall that \(\frac{a^m}{a^n} = a^{m-n}\). Apply this rule to the ratio: $$ \frac{x^{6}}{x^{6}} = x^{6-6} = x^{0} $$

Simplify \(x^0\)

Zero exponent rule states that for any non-zero base, \(a^0 = 1\). Since \(x\) is assumed to be non-zero (not mentioned otherwise), we can conclude that: $$ x^0 = 1 $$ The equivalent expression for the given expression is \(x^0\). The value of \(n\) is \(0\).

Key concepts

Properties of Exponents

Exponents have properties that simplify the way we perform calculations involving powers. These rules are vital in algebra and can make complex-looking expressions much easier to handle. Let's explore how these properties can be applied:

• Power of a Power Rule: When a power is raised to another power, you multiply the exponents. So for any number such as \((a^m)^n\), this simplifies directly to \(a^{mn}\).
• Power of a Product Rule: When multiplying two powers with the same base, you add the exponents. Thus \(a^m \times a^n = a^{m+n}\).

These rules help us to manage and reduce expressions quickly and effectively. In our exercise, the power of a power rule is specifically used to simplify both the numerator and the denominator of the expression.

Quotient Rule for Exponents

The quotient rule is another essential property that makes simplifying expressions involving division straightforward. The rule states that when you divide two powers with the same base, you subtract the exponents. The formula is:\[ \frac{a^m}{a^n} = a^{m-n} \]This rule is quite handy when we deal with expressions where both the numerator and the denominator have the same base. In the example given, \(x^6 / x^6\) becomes \(x^{6-6}\), resulting in \(x^0\). This subtraction step is crucial, providing a simple method to reduce any such fractional division of powers.

Zero Exponent Rule

The zero exponent rule states a very interesting fact about exponents: any non-zero number raised to the power of zero is equal to one. So, for any base \(a\), provided \(aeq0\), the following is always true:\[ a^0 = 1 \]This rule might seem a bit counterintuitive at first, but it is derived from the need for consistent application of other exponent rules. For instance, using the quotient rule, if we have an equal number of factors in the numerator and denominator, they cancel out, leaving the base raised to a power of zero, which is equivalent to one.In the exercise, \(x^0\) simplifies to 1, applying this fundamental rule. It's a simple yet powerful concept that ensures the consistency of calculations involving exponents.