Question

Simplify (a) \(x^{2} x^{5} x^{9}\), (b) \(\frac{x^{9}}{x^{4}}\), (c) \(\frac{x^{4}}{x^{5}}\).

Short answer

Question: Simplify the following expressions: (a) \(x^{2} x^{5} x^{9}\), (b) \(\frac{x^{9}}{x^{4}}\), and (c) \(\frac{x^{4}}{x^{5}}\). Answer: The simplified expressions are: (a) \(x^{16}\), (b) \(x^{5}\), and (c) \(x^{-1}\).

Step-by-step solution

(a) Simplify \(x^{2} x^{5} x^{9}\)

To simplify the expression, we will use the product rule. Combine all the exponents by adding them together. \(x^{2} x^{5} x^{9} = x^{(2+5+9)}\) Now, add the exponents: \(x^{(2+5+9)} = x^{16}\) So, the simplified expression is \(x^{16}\).

(b) Simplify \(\frac{x^{9}}{x^{4}}\)

In order to simplify the expression, we will apply the quotient rule. Subtract the exponent of the denominator from the exponent of the numerator. \(\frac{x^{9}}{x^{4}} = x^{(9-4)}\) Now, subtract the exponents: \(x^{(9-4)} = x^{5}\) So, the simplified expression is \(x^{5}\).

(c) Simplify \(\frac{x^{4}}{x^{5}}\)

To simplify this expression, we will use the quotient rule as well. Subtract the exponent of the denominator from the exponent of the numerator. \(\frac{x^{4}}{x^{5}} = x^{(4-5)}\) Subtract the exponents: \(x^{(4-5)} = x^{-1}\) So, the simplified expression is \(x^{-1}\).

Key concepts

Product Rule in Exponents

The product rule is a handy tool when dealing with exponents. It helps in simplifying expressions where the same base is being multiplied. To use the product rule, simply add the exponents of the base being multiplied together. This rule is wonderful for making complex expressions more manageable. For instance, let's examine how it's applied:

• When you have an expression like \(x^2 \times x^5 \times x^9\), all bases are the same \(x\).
• According to the product rule, we add up the exponents: \(2 + 5 + 9 = 16\).
• The result becomes \(x^{16}\).

This rule is very straightforward once you get the hang of it and can make your calculations much simpler.

Quotient Rule in Exponents

The quotient rule is your go-to method for simplifying expressions involving division of powers with the same base. It dictates that you subtract the exponent of the denominator from that of the numerator. Let's see this rule in action:

• Take the expression \(\frac{x^9}{x^4}\).
• The bases are the same (\(x\)), so you subtract the exponents: \(9 - 4 = 5\).
• The simplified form is \(x^5\).

This is quite intuitive once you understand that division leads to decreasing the power. Don't be confused if the result sometimes turns out to be a negative exponent. That's also covered by this rule.

Simplifying Expressions with Exponents

Simplifying expressions where exponents come into play helps you work more effectively with equations and formulas. It's important to understand that both the product and quotient rules are integral in this process. Let's sum up this understanding:

• When multiplying like terms, use the product rule to add exponents.
• When dividing like terms, use the quotient rule to subtract exponents.
• Both rules help in rewriting expressions in their simplest form, making calculations or further algebraic manipulation easier.

These rules help you break down larger problems into manageable parts, smoothing out what could otherwise be a complex process. By practicing these steps, you'll find yourself mastering exponent simplifications in no time!