Question

Simplify \(\frac{25 x^9 y^4}{15 x^6 y^{12}}\). A. \(\frac{10 x^3 y}{x^2 y^3}\) B. \(\frac{10 x^3}{y^8}\) C. \(\frac{5 x^3 y}{3 x^2 y^3}\) D. \(\frac{5 x^3}{3 y^8}\)

Short answer

D. \(\frac{5 x^3}{3 y^8}\)

Step-by-step solution

Simplify the coefficients

The coefficients are 25 and 15. Simplify \(\frac{25}{15}\). Both numbers can be divided by their greatest common divisor, which is 5: \(\frac{25 \div 5}{15 \div 5} = \frac{5}{3}\).

Simplify the powers of x

For the powers of x, use the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\). Here we have \(\frac{x^9}{x^6} = x^{9-6} = x^3\).

Simplify the powers of y

Similarly, for the powers of y, use the quotient rule: \(\frac{y^4}{y^{12}} = y^{4-12} = y^{-8} = \frac{1}{y^8}\).

Combine all parts

Combine the simplified coefficients and variables: \(\frac{5}{3} \times x^3 \times \frac{1}{y^8} = \frac{5 x^3}{3 y^8}\).

Key concepts

Quotient Rule

The quotient rule is a handy tool for simplifying fractions with variables raised to powers. The rule states that \(\frac{a^m}{a^n} = a^{m - n}\). This means when you divide two powers with the same base, you subtract the exponent in the denominator from the exponent in the numerator.

For example, in the given problem, we have \(\frac{x^9}{x^6} = x^{9 - 6} = x^3\). This subtraction makes it straightforward to simplify expressions involving powers of the same base.

Here's a quick recap:

\(\frac{a^m}{a^n} = a^{m - n}\) - Subtract the exponents

Remember, this only applies when the bases are the same. This technique is not only easy but also very powerful for simplifying algebraic expressions.

Power of Variables

To understand simplifying expressions with variables, you need to grasp the laws of exponents. The two most important rules are the product rule and the quotient rule.

In our specific problem, we use the quotient rule. When simplifying \(\frac{x^9}{x^6}\), it's like saying \((x \times x \times x \times x \times x \times x \times x \times x \times x) \/ (x \times x \times x \times x \times x \times x)\). The six x's in the denominator cancel out with six x's in the numerator, leaving us with \(x^3\).

A similar concept applies to the y variables: \(\frac{y^4}{y^{12}} = y^{4-12} = y^{-8}\). Here, we end up with \(y^{-8}\), which is the same as \(\frac{1}{y^8}\). The negative exponent tells us to put the term in the denominator.

Always remember:

Positive exponents stay in the numerator.
Negative exponents move to the denominator.

Understanding these rules can make working with variables and exponents much easier to manage.

Greatest Common Divisor

The greatest common divisor (GCD) is essential for simplifying fractions with numerical coefficients. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

In the example \(\frac{25}{15}\), both 25 and 15 share a GCD of 5. This means both numbers can be divided by 5 to simplify the fraction further:

\(\frac{25}{15} = \(\frac{25 \/ 5}{15 \/ 5} = \frac{5}{3}\)\).

Finding the GCD can be done using various methods, but one of the simplest involves prime factorization or using the Euclidean algorithm.

Steps to find the GCD:

List the prime factors of each number.
Identify the common factors.
Multiply these common factors together.

The GCD helps reduce fractions to their simplest form. Applying this concept helps make complex calculations more manageable and results clearer.