Problem 33 - Algebra 1: Concepts and Skills

Question

Write the quotient in simplest form. $$\frac{3 x^{2}}{10} \div \frac{9 x^{3}}{25}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{3 x^{2}}{10} \div \frac{9 x^{3}}{25}\) is \(\frac{5}{6x}\)

Step 1: Rewrite the division as multiplication

The expression \(\frac{3 x^{2}}{10} \div \frac{9 x^{3}}{25}\) can be written as \(\frac{3 x^{2}}{10} * \frac{25}{9 x^{3}}\). As dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 2: Multiply the Fractions

Now, we multiply the numerators to get the numerator of the result and multiply the denominators to get the denominator of the result. Doing so, we get \(\frac{3 * 25 * x^{2}}{10 * 9 * x^{3}}\).

Step 3: Simplify the Fractions

Simplify the above fraction by canceling out common factors in the numerator and the denominator, we get \(\frac{75}{90}\) and \(x^{2-3}\) respectively, where we subtract the powers as per the law of exponents.

Step 4: Further Simplify the Fractions

Further simplify the fraction by dividing the numerator and the denominator by their greatest common divisor which is 15, we get \(\frac{5}{6}\). For \(x^{2-3}\) we simplify to get \(x^{-1}\), which can be expressed as \(\frac{1}{x}\)

Step 5: Combine the results

Combine the results to get the final answer. So, \(\frac{75}{90} * x^{-1}\) simplifies to \(\frac{5}{6x}\)

Key Concepts

Simplifying Rational Expressions

When faced with a rational expression, simplifying it often makes it more understandable and easier to work with. Just like simplifying a fraction, when simplifying a rational expression, the goal is to reduce it to its simplest form. This typically involves factoring polynomials and then canceling out common factors from the numerator and denominator.

Here's how you can approach this:

• Factor the numerator and denominator separately. Look for common factors, including numeric coefficients, variables, or polynomial factors.
• Cancel out any common factors that appear in both the numerator and the denominator. Canceling is valid because it's akin to dividing by the same non-zero number.
• Make sure to consider the restricted values for the variables which would make the original rational expression undefined (such as values that would make the denominator zero).

Remember, the simplified expression still represents the same value, it's just been stripped down to its essential components.

Dividing Polynomials

Dividing polynomials can seem daunting at first, but understanding how to convert a division problem into a multiplication problem simplifies the process significantly. This method is sometimes called 'multiplying by the reciprocal'. When you divide by a polynomial, it is the same as multiplying by its reciprocal—flip the fraction you're dividing by.

For instance, \

\(\frac{x^2}{y}\) divided by \

\(\frac{z}{w}\)

is the same as \

\(\frac{x^2}{y}\) multiplied by \

\(\frac{w}{z}\)

.

After the flip, you're now multiplying two polynomials. From this point:

• Distribute (if necessary) to multiply the numerators together and the denominators together.
• Look for opportunities to simplify, keeping in mind the distributive property and any common factors.
• Factor and reduce if applicable.
• Keep an eye out for complex factors which might need additional steps like polynomial division or synthetic division.

This technique transforms division of polynomials into an easier, more familiar operation.

Law of Exponents

The law of exponents is critical when simplifying algebraic expressions, particularly those involving variables raised to powers. These laws allow you to manipulate and simplify expressions without changing their values. They include important rules such as:

• The product rule: \

  \(x^a \cdot x^b = x^{a+b}\)

  , which states that when you multiply two powers with the same base, you add the exponents.
• The quotient rule: \

  \(\frac{x^a}{x^b} = x^{a-b}\)

  , which tells us that when we divide two powers with the same base, we subtract the exponents.
• The power of a power rule: \

  \(\(x^a\)^b = x^{a\cdot b}\)

  , indicating that when raising a power to another power, you multiply the exponents.

Remember, when you come across negative exponents, \

\(x^{-n} = \frac{1}{x^n}\)

, this implies that a negative exponent represents the reciprocal of the positive exponent. The understanding of these laws streamlines the process of simplification and lays the foundation for mastering algebra.