Problem 58 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\frac{10 x^{4} y^{9}}{30 x^{12} y^{-3}}$$

Step-by-Step Solution

Answer

The simplified exponential expression is \( \frac{1/3 y^{12}}{x^{8}}\)

Step 1: Simplify the Coefficients

Divide the coefficient 10 in the numerator by the coefficient 30 in the denominator. The coefficient simplifies to \(\frac{1}{3}\)

Step 2: Simplify the x Terms

Use the rule that when you're dividing terms with the same base, you subtract the exponents. So \(x^{4}\) divided by \(x^{12}\) equals \(x^{4-12} = x^{-8}\). This denotes that \(x^{-8}\) is in the denominator.

Step 3: Simplify the y Terms

Similarly, for \(y^{9}\) divided by \(y^{-3}\), subtract the exponents \(9 - (-3) = 12\), which gives \(y^{12}\). The term \(y^{12}\) remains in the numerator.

Step 4: Construct the Simplified Expression

Combine the simplified coefficients, x terms and y terms together to get the final simplified expression, \( \frac{1/3 y^{12}}{x^{8}}\)

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking them down into their simplest form. This is a key skill in algebraic operations. Think of it as reducing clutter in a complex equation. When you simplify, you're using mathematical properties to make expressions more manageable and easier to work with.

For example, when simplifying the expression \(\frac{10 x^{4} y^{9}}{30 x^{12} y^{-3}}\), we separate the coefficients from the variables. Start by simplifying each part using basic mathematical principles. By dividing coefficients and applying rules of exponents to the variables, you arrive at a streamlined version of the expression.

Exponents Subtraction

When dealing with exponents, a crucial rule is the subtraction of exponents during division. This applies when the bases are identical. In the given expression \(\frac{10 x^{4} y^{9}}{30 x^{12} y^{-3}}\), notice that each term in the numerator has a corresponding term in the denominator with the same base, like \(x\) and \(y\).

For the \(x\) terms, use the exponent subtraction rule: \(x^{4} \div x^{12} = x^{4-12} = x^{-8}\). Negative exponents can be rewritten as a fraction. Therefore, \(x^{-8}\) appears as \(\frac{1}{x^8}\).

For the \(y\) terms, \(y^{9} \div y^{-3} = y^{9 - (-3)} = y^{12}\). Here, adding the exponents transforms their operation, since subtracting a negative becomes addition.

Numerator and Denominator

In any fraction, the top part is the numerator and the bottom part is the denominator. Simplifying these components is essential in reducing expressions. Understanding the roles these parts play helps in managing complex fractions.

When simplifying \(\frac{10 x^{4} y^{9}}{30 x^{12} y^{-3}}\), start by simplifying the coefficients which are 10 and 30. \(\frac{10}{30}\) reduces to \(\frac{1}{3}\). Then, address the variables separately, as we did with \(x^{4}\) and \(y^{9}\).

The numerator now is \(1 \cdot y^{12}\), and the denominator is \(3 \cdot x^{8}\), resulting in \(\frac{1/3 y^{12}}{x^{8}}\), showing a balance and simplicity between the numerator and denominator.

Algebraic Expressions

Algebraic expressions can include variables, numbers, and operation symbols. Simplification often requires leveraging properties of numbers and rules governing exponents.

Take \(\frac{10 x^{4} y^{9}}{30 x^{12} y^{-3}}\) as an example. This expression combines coefficients and variables, making the exponent rules applicable to both.

Understanding each component, whether it's a standalone number or a power of a variable, helps you see how these parts interact. During simplification, we treat the algebraic expression by isolating and simplifying one part at a time—not only making complex expressions easier but also honing your overall algebraic skills.