Problem 45 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \frac{16 x^{5} y^{-8}}{x^{7} y^{4}} \cdot\left(\frac{x^{3} y^{2}}{8 x y}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^6}{256y^8} \).

Step 1: Simplify the first fraction

To simplify the first fraction, subtract the exponents of like terms from the numerator and denominator. Hence, \( \frac{16 x^{5} y^{-8}}{x^{7} y^{4}} \) simplifies to \( 16x^{-2}y^{-12} \) or \( \frac{16}{x^2 y^{12}} \).

Step 2: Simplify the second expression

Raise the entire inner expression to the power of 4, multiplying all exponents inside the brackets by 4. Hence, \( \left( \frac{x^3 y^2}{8x y} \right)^4 \) is equal to \( \frac{x^{12} y^{8}}{4096x^4 y^4} \). Reduce to \( \frac{x^8 y^4}{4096} \).

Step 3: Multiplication of the parts obtained

Multiply the expressions obtained in step 1 and step 2: \( \frac{16}{x^2 y^{12}} \cdot \frac{x^8 y^4}{4096} = \frac{16x^8 y^4}{4096x^2 y^{12}} \). Simplify by subtracting the exponents to get \( \frac{x^6}{256y^8} \).

Key Concepts

Understanding Exponents

Exponents are an essential part of mathematics that show how many times a number is multiplied by itself. For example, in the expression \(x^5\), the base is \(x\) and the exponent is 5. This means \(x\) is multiplied by itself 5 times: \(x \cdot x \cdot x \cdot x \cdot x \).

When simplifying expressions involving exponents, it's important to remember a few key rules:

• Multiplying with the same base: Add the exponents. For example, \(x^a \cdot x^b = x^{a+b}\).
• Dividing with the same base: Subtract the exponents. For instance, \(\frac{x^a}{x^b} = x^{a-b}\).
• Raising a power to a power: Multiply the exponents. As in \((x^a)^b = x^{a \cdot b}\).

Understanding these basic rules will help you simplify complex expressions efficiently.

Working with Negative Exponents

Negative exponents can initially be confusing, but they simply represent the reciprocal of the base raised to the opposite positive exponent. For instance, \(x^{-n}\) is equal to \(\frac{1}{x^n}\).

Let's look at a few examples to clarify this idea:

• \(x^{-2}\) becomes \(\frac{1}{x^2}\).
• In the expression \(\frac{16 x^{5} y^{-8}}{x^{7} y^{4}}\), the \(y^{-8}\) in the numerator moves to the denominator as \(y^8\) when simplified.

By understanding negative exponents, you can transform expressions easily and eliminate any negative signs by flipping them over a fraction line.

Mastering Fraction Simplification

Simplifying fractions involves reducing them to their simplest form by dividing the numerator and the denominator by their greatest common factor. It's no different when applied to fractional expressions involving variables and exponents.

In such expressions, remember to:

• Inspect each part separately: First, handle any expressions with exponents, then simplify the numbers by dividing them.
• Use exponential laws: Apply subtraction of exponents for division and multiplication of exponents for powering expressions.
• Reduce constants where possible: Simplify numerical coefficients by finding common factors.

In our problem, the expression \(\frac{16x^8y^4}{4096x^2y^{12}}\) simplifies by dividing 16 by 4096, resulting in \(\frac{1}{256}\), and subtracting exponents in the terms to produce the final form.