Problem 62 - Algebra 2

Question

Simplify. Assume that no variable equals \(0 .\) $$ \left(\frac{3 a b^{2}}{6 a^{2} b}\right)^{2} $$

Step-by-Step Solution

Answer

Simplified form is \(\frac{b^2}{4a^2}\).

Step 1: Simplify the Fraction

Look at the expression inside the parentheses: \( \frac{3ab^{2}}{6a^{2}b} \). Let's simplify it by cancelling common factors. The numerator is \(3a b^2\) and the denominator is \(6a^2 b\). Start by cancelling \(a\) in the numerator and \(a^2\) in the denominator. This leaves \(a^0\) in numerator (which is essentially 1), and \(a\) in the denominator. Similarly, cancel \(b\) in the numerator and denominator, leaving \(b\) in the numerator. The numbers 3 and 6 can also be simplified by dividing both by 3, resulting in \(\frac{1}{2}\). The result of simplifying is \(\frac{b}{2a}\).

Step 2: Square the Simplified Fraction

Now that we have simplified the expression inside the parentheses to \(\frac{b}{2a}\), let's square it as indicated by the expression: \( \left(\frac{b}{2a}\right)^{2} \). When squaring a fraction, square both the numerator and the denominator. So: \((b)^2 = b^2\) and \((2a)^2 = 4a^2\). This gives us the expression: \(\frac{b^2}{4a^2}\).

Step 3: Verify Assumptions About Zeros

Lastly, verify the given assumption that no variable equals 0. We can confirm that in this context, since no division by zero occurs, given \(aeq0\) and \(beq0\). The expression is valid and does not need further alterations.

Key Concepts

Fraction Exponents

Fraction exponents represent a powerful tool in algebraic expressions that allow us to rewrite radicals and simplify expressions more effectively. When you see an exponent expressed as a fraction, it's actually a combination of an integer exponent and a root. For instance, we'll denote a base as \(x\) raised to a fractional power as \(x^{m/n}\). Here:

• The numerator \(m\) indicates the power to which the base \(x\) is raised.
• The denominator \(n\) signifies the root that should be taken.

Hence, \(x^{m/n}\) is equivalent to taking the \(n\)-th root of \(x\) and then raising the result to the \(m\)-th power (or vice versa). This conversion can be very helpful when simplifying complex expressions, particularly when working with roots and powers.
Consider an example where this might apply: simplifying \((x^{1/2})^4\). Here, you first take the square root of \(x\), then raise the result to the 4th power, simplifying it to \(x^2\), which shows how fraction exponents allow for clean and efficient calculations.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations to create meaningful mathematical statements. Understanding these expressions is crucial as they form the foundation for most algebraic calculations. Variables are symbols, usually letters, that represent numbers, and they can vary or change within the context of a study.
In algebraic expressions, you'll often encounter terms separated by addition or subtraction. Each term can include coefficients (the numbers in front of variables) and exponents, as shown in the original exercise: \(\frac{3ab^2}{6a^2b}\). Here:

• \(3ab^2\) is the numerator, where \(3\) is a coefficient, \(a\) is a variable, and \(b^2\) indicates \(b\) raised to the second power.
• \(6a^2b\) is the denominator, with \(6\) as the coefficient.

Simplifying such expressions involves combining like terms, cancelling common factors, and reducing the expression to its simplest form. The simplification process helps to streamline calculations and solve complex algebraic problems more efficiently.

Exponent Properties

Exponent properties form the rules that govern how we handle powers of numbers and variables. These properties simplify and manage expressions involving exponents, making calculations more straightforward. Here are several key exponent properties used often:

• Product of Powers Property: When multiplying two powers with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Property: When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator: \(a^m / a^n = a^{m-n}\).
• Power of a Power Property: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• Zero Exponent Property: Any base with an exponent of zero equals 1 (except for \(0\)): \(a^0 = 1\).
• Negative Exponent Property: A negative exponent signifies the reciprocal of the base raised to the opposite positive exponent: \(a^{-n} = 1/a^n\).

Understanding and applying these properties correctly, such as when simplifying \((\frac{b}{2a})^2\) to \(\frac{b^2}{4a^2}\), enables efficient resolution of algebraic expressions. These properties are invaluable tools, particularly when faced with exercises involving multiple exponents and complex variable combinations.